7 Operator Convexity and Jensen Inequalities
This chapter collects the operator convexity/concavity results, trace inequalities, the operator Jensen inequality for positive maps, and the Lieb concavity theorem. These are standard results in matrix analysis ( [ Bha97 , Chapter V ] ; [ Wol12 , Theorems 5.12 and 5.13 ] ; Hansen–Pedersen’s Jensen inequality [ HP82 ] ; and Lieb’s concavity theorem [ Lie73 ] ). The operator Jensen inequalities for real powers \(x\mapsto x^p\) on the Loewner order are obtained from the Loewner integral representation of the power function. Lieb’s concavity theorem is derived from the operator integral representation of the fractional product together with the vectorization isometry, which turns the trace functional into a quadratic form of a positive-semidefinite matrix.
7.1 Schwarz inequalities for positive maps
The following consequences of positivity and the Kadison–Schwarz inequality concern normal, subnormal, and commuting-dominant operators. They provide matrix-order tools used independently of the Fundamental Theorem.
7.1.1 Schwarz inequality for normal operators
Let \(E^*(X)=\sum _i K_i^\dagger X K_i\) be the adjoint Kraus map of a trace-preserving Kraus family, and let \(A\in M_{D}(\mathbb {C})\) be normal. Then the Schwarz gap
is positive semidefinite. Equivalently,
This is the completely positive case of [ Wol12 , Proposition 5.1 ] .
Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive linear map with \(T(\mathbb {1})\le \mathbb {1}\). If \(A\in M_{D}(\mathbb {C})\) is normal, then \(T(A^\dagger A)-T(A^\dagger )T(A)\ge 0\). This is [ Wol12 , Proposition 5.1 ] .
Since \(A\) is normal, diagonalize \(A=UDU^\dagger \) with \(D\) diagonal and \(U\) unitary. The spectral projectors of \(A\) generate a commutative \(*\)-subalgebra. The restriction of \(T\) to this subalgebra is completely positive, so the Kadison–Schwarz inequality for completely positive subunital maps gives \(T(A^\dagger )T(A)\le T(A^\dagger A)\).
The proof follows the positivity-on-abelian argument of [ Wol12 , Proposition 1.6 ] : a positive map restricted to the commutative \(*\)-subalgebra generated by the spectral projectors of \(A\) satisfies the Schwarz inequality directly, without invoking the Arveson extension theorem.
7.1.2 Schwarz inequality for subnormal and commuting-dominant operators
Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive linear map with \(T(\mathbb {1})\le \mathbb {1}\). If \(A\in M_{D}(\mathbb {C})\) is subnormal (i.e. there exists a normal operator on a larger space whose compression to \(\mathbb {C}^D\) is \(A\)), then \(T(A^\dagger A)-T(A^\dagger )T(A)\ge 0\). This is [ Wol12 , Theorem 5.5 ] .
Embed \(A\) as the northwest corner of a normal operator \(N\) on \(\mathbb {C}^D\oplus \mathbb {C}^E\). Extend \(T\) to a positive subunital map \(\widetilde{T}\) on the larger space, acting as \(T\) on the northwest block and as zero elsewhere. Apply Theorem 7.1.1.2 to \(\widetilde{T}\) and \(N\), then compress the resulting inequality back to \(\mathbb {C}^D\).
Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive linear map with \(T(\mathbb {1})\le \mathbb {1}\). If \(A\in M_{D}(\mathbb {C})\) admits a positive semidefinite dominant \(B\ge 0\) with \(A^\dagger A\le B\) and \(B\) commuting with \(A\) (\(BA=AB\)), then
This is [ Wol12 , Theorem 5.6 ] .
First take \(B{\gt}0\). Since \(B\) commutes with \(A\) and \(A^\dagger A\le B\), also \(AA^\dagger \le B\), so \(D_R=(B-A^\dagger A)^{1/2}\) and \(D_L=(B-AA^\dagger )^{1/2}\) are positive semidefinite. The block matrix
is normal. As in the subnormal case, extend \(T\) to a positive subunital map on \(M_{2D}(\mathbb {C})\), apply the normal-operator Schwarz inequality (Theorem 7.1.1.2) to that extension and \(N\), and compress back to \(\mathbb {C}^D\). Since \(N\) is normal, both \(T(A^\dagger )T(A)\le T(B)\) and \(T(A)T(A^\dagger )\le T(B)\) follow. For general \(B\ge 0\), apply this to \(B_\varepsilon =B+\varepsilon \mathbb {1}{\gt}0\) and let \(\varepsilon \to 0\).
Let \(E^*\) be the adjoint of a trace-preserving Kraus map. If \(A\in M_{D}(\mathbb {C})\) admits a positive semidefinite dominant \(B\) commuting with \(A\) and satisfying \(A^\dagger A\le B\), then
Apply Theorem 7.1.2.2 to the adjoint Kraus map, which is unital under the trace-preservation hypothesis.
7.2 Diagonal Jensen inequality
Let \(f:\mathbb {R}\to \mathbb {R}\) be convex on \([0,\infty )\), let \(A\) be a positive semidefinite matrix on a finite-dimensional space indexed by \(n\), and let \(v\in \mathbb {C}^n\) satisfy \(\langle v,v\rangle =1\). Then
where \(f(A)\) is defined by the Hermitian continuous functional calculus.
Write \(A=U\operatorname{diag}(\mu )U^\dagger \) by the spectral theorem and set \(w=U^\dagger v\). Since \(U\) is unitary, \(p_i:=\lvert w_i\rvert ^2\) satisfies \(\sum _i p_i=\lVert v\rVert ^2=1\), so \((p_i)\) is a probability distribution on \(n\). The eigenvalues \(\mu _i\) lie in \([0,\infty )\) because \(A\) is positive semidefinite. A direct computation yields
so the scalar Jensen inequality applied to the weights \((p_i)\) and points \((\mu _i)\) gives the conclusion. See [ Bha97 , Chapter V ] .
7.3 Trace concavity and convexity of matrix powers
For \(p\in [0,1]\), PSD matrices \(A_1,A_2\), and \(t\in [0,1]\),
This follows from operator concavity of \(x\mapsto x^p\) composed with trace monotonicity on the Loewner order.
Let \(A=tA_1+(1-t)A_2\), and let \(\{ \psi _j\} \) be an orthonormal eigenbasis of \(A\) with eigenvalues \(\mu _j\ge 0\). A direct computation gives
Scalar concavity of \(x\mapsto x^p\) on \([0,\infty )\) yields \(\mu _j^p\ge t\, a_j^p+(1-t)b_j^p\). Applying Lemma 7.2.1 to the convex function \(x\mapsto -x^p\) and multiplying by \(-1\) gives
Summing over \(j\) and using \(\sum _j\operatorname{Re}\langle \psi _j,M\psi _j\rangle =\operatorname{Re}\operatorname{tr}M\) yields the conclusion. See [ Bha97 , Chapter V ] .
For \(p\in [1,2]\), PSD matrices \(A_1,A_2\), and \(t\in [0,1]\),
This follows from operator convexity of \(x\mapsto x^p\) composed with trace monotonicity on the Loewner order.
Use the eigenbasis reduction from Theorem 7.3.1, with scalar convexity rather than concavity of \(x\mapsto x^p\) on \([0,\infty )\) for \(p\ge 1\) and the convex form of the diagonal Jensen inequality.
7.4 Operator convexity of real powers
For elements \(a,b\) of a unital C\({}^\ast \)-algebra with \(0\le a,b\), an exponent \(p\in [1,2]\), and \(t\in [0,1]\), the map \(a\mapsto a^p\) is convex for the Loewner order on the positive cone:
This is the convex counterpart of the operator concavity of \(x\mapsto x^p\) for exponents \(p\in [0,1]\).
For \(p\in (1,2)\) one uses the integral representation \(a^p=\int _0^\infty f_{p,s}(a)\, d\mu (s)\) of Carlen [ Car10 , Lemma 2.8 ] , whose integrand \(f_{p,s}(x)=s^{p-1}(s^{-1}x+s(s+x)^{-1}-1)\) is operator convex: it decomposes into a linear term, a non-negative multiple of the operator convex resolvent \(x\mapsto (s\cdot 1+x)^{-1}\), and a constant. Since \(f_{p,s}(ta+(1-t)b)\le t\, f_{p,s}(a)+(1-t)f_{p,s}(b)\) for every \(s{\gt}0\), integrating this pointwise Loewner inequality against \(\mu \) gives
The endpoint \(p=1\) is the identity, and \(p=2\) is \(a\mapsto a^2\), operator convex because \(t\, a^2+(1-t)b^2-(ta+(1-t)b)^2=t(1-t)(a-b)^2\ge 0\). See [ Bha97 , Chapter V ] .
7.5 Operator Jensen inequality for positive maps
For a finite family of matrices \(\{ C_i\} _{i\in \iota }\) and a defect block \(S\), one forms an isometric dilation whose rows are the adjoints of the matrices \(C_i\) together with the adjoint of \(S\). One also forms the scalar block-diagonal matrix whose \(\iota \)-blocks carry prescribed weights and whose defect block carries a single scalar.
Let \(Y\) be positive definite and let \(W\) be an isometry. Then
Apply the Schur-complement criterion to the block matrix \(\require{mathtools}\begin{psmallmatrix} Y^{-1}& W\\ W^\dagger & W^\dagger YW\end{psmallmatrix}\). The lower-right Schur complement is zero, hence positive semidefinite, and compressing the resulting upper-left Schur complement by \(W\) gives (3).
If the defect block satisfies
then the dilation is an isometry, and compressing the weighted scalar block-diagonal matrix gives \(\sum _{i\in \iota }w_iC_iC_i^\dagger +tSS^\dagger \). The defect relation can also be written as \(\sum _{i\in \iota }C_iC_i^\dagger +SS^\dagger =\mathbb {1}\). Strict positivity of all weights together with \(t{\gt}0\) implies positive definiteness of the scalar block-diagonal matrix. If all weights are nonzero and the defect scalar is nonzero, the inverse diagonal has reciprocal weights, and compressing this inverse gives \(\sum _{i\in \iota }w_i^{-1}C_iC_i^\dagger +t^{-1}SS^\dagger \).
Expand the block matrix multiplication entrywise. The defect identity (4) gives the isometry relation after summing the \(\iota \)-blocks and the defect block, and the weighted compression identity follows from the same expansion with the scalar diagonal inserted. Positive definiteness uses strict positivity of every diagonal entry, while the inverse formula uses the nonvanishing of every diagonal entry.
Let \(w_i\ge 0\) and \(t{\gt}0\). If the defect block satisfies \(SS^\dagger =\mathbb {1}-\sum _{i\in \iota }C_iC_i^\dagger \), then
Let \(A\ge 0\) be a finite matrix. There are positive semidefinite matrices \(P_i\) and non-negative real numbers \(\lambda _i\) such that
Diagonalize the Hermitian matrix \(A\) by a unitary matrix. The matrices \(P_i\) are the conjugates of the rank-one diagonal matrix units, their sum is the identity, and positive semidefiniteness of \(A\) gives \(\lambda _i\ge 0\).
Let \(T\) be a positive subunital map, let \(A\ge 0\), and let \(t{\gt}0\). Then
Use the spectral resolution \(A=\sum _i\lambda _iP_i\). The matrices \(T(P_i)\) are positive semidefinite and sum to \(T(\mathbb {1})\le \mathbb {1}\). Write each \(T(P_i)\) as \(C_iC_i^\dagger \) using the positive square root, and write the defect \(\mathbb {1}-\sum _iT(P_i)\) as \(SS^\dagger \). The finite-POVM resolvent inequality gives the desired estimate after substituting the spectral formula for \((A+t\mathbb {1})^{-1}\).
Let \(T\) be a positive subunital map, let \(A\ge 0\), let \(p\in (0,1)\), and let \(t{\gt}0\). Then
Rewrite the continuous functional calculus in resolvent form as \(f_{p,t}(X)=t^{p-1}\mathbb {1}-t^p(t\mathbb {1}+X)^{-1}\). After applying the linear map \(T\), the difference \(f_{p,t}(T(A))-T(f_{p,t}(A))\) is exactly \(t^p\) times the positive matrix supplied by the positive-map resolvent inequality.
Let \(T\) be a positive subunital map, let \(A\ge 0\), let \(p\in (1,2)\), and let \(t{\gt}0\). Then
The inequality is reversed relative to the concave integrand, since \(g_{p,t}\) is operator convex.
Rewrite the continuous functional calculus in resolvent form as \(g_{p,t}(X)=t^{p-2}X+t^p(t\mathbb {1}+X)^{-1}-t^{p-1}\mathbb {1}\). Under the linear map \(T\), the leading \(t^{p-2}\) terms cancel and the constant \(t^{p-1}\) terms combine, so the difference \(T(g_{p,t}(A))-g_{p,t}(T(A))\) is exactly \(t^p\) times the positive matrix supplied by the positive-map resolvent inequality, using \(t^pt^{-1}=t^{p-1}\).
Let \(f\) be a function from a measure space to a finite matrix algebra. If \(f(x)\ge 0\) for almost every \(x\), then \(\int f(x)\, d\mu (x)\ge 0\).
The Loewner order has closed positive cone in finite dimension. Hence the ordered Bochner integral preserves almost-everywhere nonnegativity. The corresponding almost-everywhere monotonicity statement is the general ordered Bochner integral theorem.
For a positive subunital map \(T\), a positive semidefinite matrix \(A\), and \(p\in [0,1]\), one has \(T(A^p)\le (T(A))^p\). This is [ Wol12 , Theorem 5.13 ] for concave \(f(x)=x^p\).
For \(p=0\) the assertion is the subunitality condition \(T(\mathbb {1})\le \mathbb {1}\). For \(p=1\) it is an equality. For \(0{\lt}p{\lt}1\), use Carlen’s integral representation of \(A^p\) by the functions \(f_{p,t}(x)=t^p(t^{-1}-(t+x)^{-1})\). The pointwise inequality \(T(f_{p,t}(A))\le f_{p,t}(T(A))\) is Lemma 7.5.7. Integrating this inequality with respect to the representing measure gives the result by Lemma 7.5.9 and linearity of the Bochner integral under \(T\).
For a positive subunital map \(T\), a positive semidefinite matrix \(A\), and \(p\in [0,1]\), the inequality \(T(A^p)\le (T(A))^p\) holds.
Direct application of Theorem 7.5.10.
For a positive subunital map \(T\), a positive semidefinite matrix \(A\), and \(p\in [1,2]\), one has \((T(A))^p\le T(A^p)\). This is [ Wol12 , Theorem 5.11 ] for the subunital operator-convex case \(f(x)=x^p\).
For \(p=1\) the assertion is an equality. For \(1{\lt}p{\lt}2\), use Carlen’s integral representation of \(A^p\) by the convex integrands \(g_{p,t}(x)=t^{p-2}x+t^p(t+x)^{-1}-t^{p-1}\). The pointwise inequality \(g_{p,t}(T(A))\le T(g_{p,t}(A))\) is Lemma 7.5.8. Integrating this inequality with respect to the representing measure and using Lemma 7.5.9 together with linearity of the Bochner integral under \(T\) gives the inequality for \(1{\lt}p{\lt}2\). The endpoint \(p=2\) follows by taking the limit \(q\to 2^-\) in \((T(A))^q\le T(A^q)\): since \(M^q\to M^2\) in the operator norm as \(q\to 2^-\) for any positive semidefinite \(M\), and the Loewner order is closed, this gives \((T(A))^2\le T(A^2)\).
For a positive subunital map \(T\), a positive semidefinite matrix \(A\), and \(p\in [1,2]\), the inequality \((T(A))^p\le T(A^p)\) holds.
Direct application of Theorem 7.5.12.
For a positive unital map \(T\) and positive-definite \(A\), one has \(T(\log A)\le \log (T(A))\). This is the corrected logarithmic specialization associated with [ Wol12 , Corollary 5.2(3) ] . The printed common hypothesis \(T(\mathbb {1})\le \mathbb {1}\) is insufficient: in dimension one, for \(0{\lt}c{\lt}1\), the positive subunital map \(T(x)=cx\) and \(A=1\) would give \(0=T(\log A)\le \log (T(A))=\log c\). The corrected theorem instead assumes \(T(\mathbb {1})=\mathbb {1}\); see [ con26k ] .
Apply Theorem 7.5.10 to \(0{\lt}p{\lt}1\) and rewrite it as an inequality for \((A^p-\mathbb {1})/p\). The continuous functional calculus gives \((A^p-\mathbb {1})/p\to \log A\) as \(p\to 0^+\), and the same limit holds for \(T(A)\). Since the Loewner order graph is closed in finite dimension, the limiting inequality is \(T(\log A)\le \log (T(A))\).
For a positive unital map \(T\) and a positive-definite matrix \(A\), the inequality \(T(\log A)\le \log (T(A))\) holds.
Direct application of Theorem 7.5.14.
7.5.1 Special functions and the general inequalities
The power and logarithm inequalities above are special cases of the function-theoretic results in Wolf’s Section 5.4. For an operator-convex function \(f\) with \(f(0)\le 0\) and a positive map \(T\) satisfying \(T(\mathbb {1})\le \mathbb {1}\), Wolf’s Theorems 5.10–5.11 give \(f(T(A))\le T(f(A))\); Theorem 5.12 removes the condition \(f(0)\le 0\) when \(T\) is unital. For an operator-monotone function \(f\) with \(f(0)\ge 0\), Theorem 5.13 gives the reverse inequality \(T(f(A))\le f(T(A))\). These assertions concern arbitrary functions in the respective classes, whereas Corollary 5.2 specializes them to powers and the logarithm. The distinction is recorded in [ con26g ] ; the correction to the logarithmic clause is recorded separately in [ con26k ] .
7.6 Source boundary for Wolf’s unitary comparison
Wolf’s Proposition 5.3 claims that, for arbitrary Hermitian \(A,B\), there is one unitary \(U\) such that
for every scalar non-decreasing function \(f\) on an interval containing both spectra. The statement is false without the hypothesis \(A\le B\): in dimension one, \(A=[1]\), \(B=[0]\), and \(f(x)=x\) would give \(1\le 0\).
The corrected proposition assumes \(A\le B\). This is precisely the hypothesis of Weyl monotonicity in the immediately preceding Equation (5.56), which gives \(\lambda _j^\downarrow (A)\le \lambda _j^\downarrow (B)\) for each decreasingly ordered eigenvalue. Applying \(f\) entrywise and aligning the two ordered eigenbases proves the unitary comparison, with the same unitary for every \(f\). The exact correction and source proof route are recorded in [ con26d ] .
7.7 Lieb concavity theorem
The Lieb concavity theorem rests on a scalar integral identity that expresses the geometric interpolation \(a^s b^{1-s}\) as a positive average of the elementary fractions \(ab/(a+tb)\). This is the analytic core of the integral representation of \(A^s B^{1-s}\).
For \(s\in (0,1)\),
This is the value \(B(s,1-s)\) of the Beta function, which equals \(\Gamma (s)\Gamma (1-s)=\pi /\sin (\pi s)\) by Euler’s reflection formula. The substitution \(x=u/(1+u)\) carries the half-line onto the unit interval and turns the integrand into the Beta integrand \(x^{s-1}(1-x)^{-s}\).
For \(a,b{\gt}0\) and \(s\in (0,1)\), the function \(t\mapsto t^{s-1} \dfrac {ab}{a+tb}\) is integrable on \((0,\infty )\).
The integrand is \(O(t^{s-1})\) as \(t\to 0^+\) and \(O(t^{s-2})\) as \(t\to \infty \), so it has integrable comparison functions on each tail.
For \(a,b{\gt}0\) and \(s\in (0,1)\),
The scaling substitution \(t=(a/b)\, u\) turns the integrand into \(a^s b^{1-s}\, u^{s-1}/(1+u)\), whose half-line integral is the reflection integral \(\pi /\sin (\pi s)\) of Lemma 7.7.1.
For \(a,b{\gt}0\) and \(s\in (0,1)\),
By Lemma 7.7.3, the integral equals \(a^s b^{1-s} \pi /\sin (\pi s)\), so
For positive diagonal matrices \(M\) and \(N\) and \(s\in (0,1)\),
Both sides are diagonal, so the identity holds entry by entry; on each diagonal entry it is the scalar Lieb integral identity (Lemma 7.7.4).
For positive-definite matrices \(A\), \(B\) and \(s\in (0,1)\),
where \(A\otimes \mathbb {1}\) and \(\mathbb {1}\otimes B^\top \) are regarded as operators on the Kronecker model space \(\mathbb {C}^{D\times D}\).
The pair \((A\otimes \mathbb {1},\mathbb {1}\otimes B^\top )\) is simultaneously diagonalized by \(V=U_A\otimes U_{B^\top }\) (eigenbases of \(A\) and \(B^\top \)). In that basis the pair becomes a commuting pair of positive diagonal matrices, and the identity follows from the diagonal case (Lemma 7.7.5).
For positive-definite matrices \(X_1,X_2,Y_1,Y_2\) and \(\theta \in [0,1]\), the parallel sum \((X,Y)\mapsto X(X+Y)^{-1}Y\) is jointly concave in the Loewner order:
where \(X_\theta =\theta X_1+(1-\theta )X_2\) and \(Y_\theta =\theta Y_1+(1-\theta )Y_2\).
Write \(Z=X(X+Y)^{-1}Y\). The identity \(X(X+Y)^{-1}(X+Y)=X\) gives \((X-Z)-X(X+Y)^{-1}X=0\), so the block matrix \(\require{mathtools}\begin{psmallmatrix} X-Z & X\\ X & X+Y\end{psmallmatrix}\) is positive semidefinite (its Schur complement against \(X+Y\) vanishes). Setting \(Z_i=X_i(X_i+Y_i)^{-1}Y_i\) and \(Z_\theta =\theta Z_1+(1-\theta )Z_2\), the convex combination of the two block matrices is \(\require{mathtools}\begin{psmallmatrix} X_\theta -Z_\theta & X_\theta \\ X_\theta & X_\theta +Y_\theta \end{psmallmatrix}\), which is positive semidefinite, and its Schur complement against \(X_\theta +Y_\theta \) yields \(X_\theta (X_\theta +Y_\theta )^{-1}Y_\theta -Z_\theta \ge 0\), which is the claimed inequality (Ando, 1979).
For \(t{\gt}0\), positive-definite \(A_1,A_2,B_1,B_2\), and \(\theta \in [0,1]\), with \(\hat{A}=A\otimes \mathbb {1}\) and \(\hat{B}=\mathbb {1}\otimes B^\top \), setting \(A_\theta =\theta A_1+(1-\theta )A_2\) and \(B_\theta =\theta B_1+(1-\theta )B_2\):
The integrand equals \(t^{-1} \hat A(\hat A+t\hat B)^{-1}(t\hat B)\), a rescaled parallel sum of \(\hat A\) and \(t\hat B\), so concavity follows from Lemma 7.7.7 and the linearity of \(A\mapsto \hat A\), \(B\mapsto \hat B\).
For positive-definite matrices \(A_1,A_2,B_1,B_2\), \(s\in (0,1)\), and \(\theta \in [0,1]\), with \(\hat{A}=A\otimes \mathbb {1}\) and \(\hat{B}=\mathbb {1}\otimes B^\top \), setting \(A_\theta =\theta A_1+(1-\theta )A_2\) and \(B_\theta =\theta B_1+(1-\theta )B_2\), the fractional product \(\hat A^s\hat B^{1-s}\) is jointly concave in the Loewner order:
By the integral representation (7), each fractional product equals the positive multiple \(\frac{\sin (\pi s)}{\pi }\) of the integral of the resolvent integrand against the non-negative weight \(t^{s-1}\) on \((0,\infty )\). For each \(t{\gt}0\), the integrand is jointly concave by Lemma 7.7.8, so the difference of the averaged point and the convex combination is, almost everywhere, the non-negative scalar \(t^{s-1}\) times a positive-semidefinite matrix. Integrating a positive-semidefinite integrand yields a positive-semidefinite matrix, and scaling by the positive constant \(\frac{\sin (\pi s)}{\pi }\) preserves the Loewner order, which gives the inequality.
For \(s\in [0,1]\), any matrix \(K\), and positive-definite matrices \(A_1,A_2,B_1,B_2\), write \(F_s(A,B):=\Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\). This map is jointly concave. For \(t\in [0,1]\), set \(A_t:=tA_1+(1-t)A_2\) and \(B_t:=tB_1+(1-t)B_2\). Then
See [ Lie73 ; And79 ] . This statement is the positive-definite, boundary-exponent (\(x+y=1\)) case of the Ando–Lieb theorem [ Wol12 , Theorem 5.15 ] , which holds for positive semidefinite \(A,B\) and all exponents \(x,y\ge 0\) with \(x+y\le 1\).
For \(s\in (0,1)\), the concavity of the fractional product of the commuting left and right multiplication operators (Lemma 7.7.9) gives a Loewner-order inequality between positive-semidefinite matrices on the model space \(\mathbb {C}^D\otimes \mathbb {C}^D\). Writing that product as \(A^s\otimes (B^\top )^{1-s}\) and pairing it with the vectorization of \(K^\top \),
expresses the trace functional as the quadratic form of the Kronecker product at that vector. Applying this positive quadratic form to the Loewner inequality transfers the concavity to the real part of the trace. The endpoints \(s=0\) and \(s=1\) reduce the varying side to a linear function of the convex-combination argument, hence hold with equality.
For \(s\in [0,1]\), any matrix \(K\), and positive-semidefinite matrices \(A_1,A_2,B_1,B_2\), the map \((A,B)\mapsto \Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\) is jointly concave, satisfying (8) of Theorem 7.7.10. This is the full Ando–Lieb theorem [ Wol12 , Theorem 5.15 ] on the boundary line \(x+y=1\) (with \(x=s\), \(y=1-s\)): the positive-definiteness restriction of Theorem 7.7.10 is lifted. The general two-exponent region is obtained in Theorem 7.7.12.
For the interior exponents \(s\in (0,1)\), replace each input \(A_i\) by the positive-definite matrix \(A_i+\varepsilon \mathbb {1}\), and likewise for the \(B_i\). Theorem 7.7.10 applies to these positive-definite matrices for every \(\varepsilon {\gt}0\), and the average of the regularized inputs is the regularization of the average. Letting \(\varepsilon \to 0^+\), each regularized power \((A_i+\varepsilon \mathbb {1})^s\) converges to \(A_i^s\), since \(x\mapsto x^s\) is continuous on the non-negative reals for \(0{\lt}s{\lt}1\), including at the origin; the trace functional is continuous, so the regularized inequality passes to the limit. The endpoints \(s=0\) and \(s=1\) are linear in the convex-combination argument and hold with equality.
Let \(x,y\ge 0\) with \(x+y\le 1\). For any matrix \(K\) and positive-semidefinite matrices \(A_1,A_2,B_1,B_2\), write \(F_{x,y}(A,B):=\Re \operatorname{tr}(K^\dagger A^x K B^y)\). This map is jointly concave. For \(t\in [0,1]\), set \(A_t:=tA_1+(1-t)A_2\) and \(B_t:=tB_1+(1-t)B_2\). Then
This is Wolf’s Theorem 5.15 in the finite-dimensional matrix setting.
Put \(r=x+y\). If \(r=0\), both exponents vanish and the functional is constant. If \(0{\lt}r\le 1\), set \(s=x/r\). Then \(x=rs\) and \(y=r(1-s)\), with \(s\in [0,1]\). The functional becomes the boundary Lieb functional applied to the powered inputs \(A^r\) and \(B^r\). Operator concavity of \(C\mapsto C^r\) gives
The boundary positive-semidefinite theorem gives concavity at the left-hand powered inputs, and monotonicity of \((P,Q)\mapsto \Re \operatorname{tr}(K^\dagger P^s K Q^{1-s})\) on positive-semidefinite arguments transports the result to the powered convex combinations.
Follows directly from Theorem 7.7.10.
For \(s\in [0,1]\), \(t\in [0,1]\), and positive-definite matrices \(A_1,A_2,B_1,B_2\), the map \((A,B)\mapsto \Re \operatorname{tr}(A^s B^{1-s})\) is jointly concave:
Obtained from Theorem 7.7.13 by taking \(K=\mathbb {1}\).
Specialize Theorem 7.7.13 at \(K=\mathbb {1}\) and simplify \(\mathbb {1}^\dagger =\mathbb {1}\) together with \(\mathbb {1}\cdot X=X\cdot \mathbb {1}=X\).
For \(s\in [0,1]\), \(t\in [0,1]\), matrix \(K\), and positive-definite matrices \(A_1,A_2,B\), the map \(A\mapsto \Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\) is concave:
Specialize Theorem 7.7.13 at \(B_1=B_2=B\), so that the convex combination collapses: \(tB+(1-t)B=B\).
For \(s\in [0,1]\), \(t\in [0,1]\), matrix \(K\), and positive-definite matrices \(A,B_1,B_2\), the map \(B\mapsto \Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\) is concave:
Specialize Theorem 7.7.13 at \(A_1=A_2=A\), so that the convex combination collapses: \(tA+(1-t)A=A\).
7.8 Resolvent monotonicity toward the Lieb concavity theorem
The integral-representation route toward the Lieb concavity theorem rests on the antitonicity of the resolvent of the commuting left- and right-multiplication superoperators. Its matrix-order foundation is the inverse-antitonicity of the Loewner order.
For positive-definite matrices \(A\) and \(B\) with \(A\le B\) in the Loewner order, the inverses satisfy \(B^{-1}\le A^{-1}\).
Compare the two Schur complements of the block matrix \(\require{mathtools}\begin{psmallmatrix} A^{-1}& \mathbb {1}\\ \mathbb {1}& B\end{psmallmatrix}\). Complementing its \((1,1)\) block gives \(B-(A^{-1})^{-1}=B-A\ge 0\), so the block matrix is positive semidefinite; complementing the \((2,2)\) block of the same matrix gives \(A^{-1}-B^{-1}\), which is therefore positive semidefinite as well.
Fix \(t{\gt}0\) and positive-definite matrices with \(A_1\le A_2\) and \(B_1\le B_2\). The resolvent of the Kronecker model \(A\otimes \mathbb {1}+t\, (\mathbb {1}\otimes B^\top )\) of the commuting left- and right-multiplication superoperators is antitone:
Each operator \(A\otimes \mathbb {1}+t\, (\mathbb {1}\otimes B^\top )\) is positive definite, being the sum of the positive-definite \(A\otimes \mathbb {1}\) and the positive-semidefinite \(t\, (\mathbb {1}\otimes B^\top )\). From \(A_1\le A_2\) and \(B_1\le B_2\) one gets \(A_1\otimes \mathbb {1}\le A_2\otimes \mathbb {1}\) and \(\mathbb {1}\otimes B_1^\top \le \mathbb {1}\otimes B_2^\top \); since \(t{\gt}0\), adding these yields
Inverse-antitonicity (Lemma 7.8.1) reverses this inequality on passing to the resolvents.
7.9 Weighted geometric means and finite mean trees
This section follows the first part of the transport section of [ Ope26b ] (Section 7, Lemma 7.1). For positive definite matrices \(A,B\) and a real parameter \(p\), the weighted geometric mean is the operation of Kubo and Ando [ KA80 ]
Finite binary trees of such means carry perturbations from the leaves to the root. Only positive definite inputs occur; the mean is never evaluated on singular matrices.
For positive definite complex matrices \(A,B\) and \(p\in \mathbb R\), set \(A\# _pB=A^{1/2}(A^{-1/2}BA^{-1/2})^pA^{1/2}\), where the real powers are taken in the functional calculus of positive definite matrices.
Let \(A,B\) be positive definite. Then \(A\# _pB\) is positive definite, \(A\# _0B=A\), \(A\# _1B=B\), and:
\((S^*AS)\# _p(S^*BS)=S^*(A\# _pB)S\) for every invertible \(S\);
\((cA)\# _p(dB)=c^{1-p}d^p\, (A\# _pB)\) for \(c,d{\gt}0\);
\(A\# _pB=B\# _{1-p}A\);
for \(0\le p\le 1\) the mean is monotone in each argument, and for every vector \(z\), \(\langle z,(A\# _pB)z\rangle \le \langle z,Az\rangle ^{1-p}\langle z,Bz\rangle ^p\);
if \(A_1,B_1\) commute with \(A_2,B_2\), then \((A_1A_2)\# _p(B_1B_2)=(A_1\# _pB_1)(A_2\# _pB_2)\), and the two factors commute.
For an invertible \(S\) put \(\widehat A=S^*AS\) and \(U=A^{1/2}S\widehat A^{-1/2}\). Then \(U^*U=I\), and \(\widehat A^{-1/2}S^*BS\widehat A^{-1/2}=U^*(A^{-1/2}BA^{-1/2})U\), so unitary covariance of the functional calculus gives the congruence rule. Scalar homogeneity follows from \((cX)^r=c^rX^r\). Congruence by \(A^{-1/2}\) reduces exchange symmetry to the identity \(C^p=C^{1/2}(C^{-1})^{1-p}C^{1/2}\) for \(C=A^{-1/2}BA^{-1/2}\). Monotonicity in \(B\) follows from operator monotonicity of \(C\mapsto C^p\) for \(0\le p\le 1\), and monotonicity in \(A\) from exchange symmetry. For the vector inequality, write \(w=A^{1/2}z\); then \(\langle z,(A\# _pB)z\rangle =\langle w,C^pw\rangle \), and Jensen’s inequality for the concave function \(x\mapsto x^p\) against the spectral measure of \(C\) in the unit vector \(w/\| w\| \) gives \(\langle w,C^pw\rangle \le \| w\| ^{2(1-p)}\langle w,Cw\rangle ^p\). For the product rule, the inverse square root of \(A_1A_2\) factors, the whitened second argument is the product of the two commuting positive definite matrices \(A_g^{-1/2}B_gA_g^{-1/2}\), and the \(p\)th power of a product of commuting positive definite matrices is the product of the powers.
A weighted binary tree is a finite rooted binary tree with a parameter \(p\in [0,1]\) at each internal vertex, whose two outgoing edges have weights \(1-p\) and \(p\). Its leaves carry labels \(j\), and the terminal weight \(w_j\) of a label is the sum, over the leaves carrying it, of the products of edge weights along the root-to-leaf paths. Assigning positive definite matrices \(A_j\) to the labels and evaluating each internal vertex by \(\# _p\) defines the root \(M\). The tree is part of the data; no associativity of matrix means is used.
A weighted binary tree is either a leaf with a label \(j\) or an internal vertex with a parameter \(p\in [0,1]\) and two subtrees. Its root on inputs \((A_j)\) is \(A_j\) at a leaf and \(M_l\# _pM_r\) at an internal vertex with subtree roots \(M_l\), \(M_r\). The terminal weights are \(\delta _j\) at a leaf and \((1-p)w^{(l)}+p\, w^{(r)}\) at an internal vertex.
The terminal weights are nonnegative and sum to one. An edge of weight zero may be omitted and the vertex with one remaining child contracted: this changes neither the root nor the terminal weights.
Induction on the tree, using \((1-p)+p=1\) and the endpoint values \(A\# _0B=A\), \(A\# _1B=B\) of Lemma 7.9.2.
Let a weighted binary tree have terminal weights \(w_j\) and positive definite inputs \(A_j\), and let \(M\) be its root. Then \(M\) is positive definite, commutes with invertible congruence, is monotone in the inputs, and satisfies \(\mathcal T(c_jA_j:j)=\bigl(\prod _jc_j^{w_j}\bigr)M\) for \(c_j{\gt}0\). A tree with constant positive definite input \(X\) has root \(X\). For every vector \(z\),
If \(P\) is positive semidefinite (in particular, an orthogonal projection) and \(A_j\ge c_jP\) with \(c_j{\gt}0\), then \(M\ge \bigl(\prod _jc_j^{w_j}\bigr)P\). Finally, if \(A_j=\prod _{g=1}^KA_{j,g}\) with positive definite factors and every factor from band \(g\) commutes with every factor from band \(g'\ne g\), then \(M=\prod _{g=1}^KM_g\), where \(M_g\) is the root of the same tree with inputs \(A_{j,g}\), and the factors \(M_g\) commute.
Positivity, congruence, monotonicity, homogeneity and the vector inequality follow by induction from Lemma 7.9.2; in the inductive step for the vector inequality the exponents combine because all terminal weights are nonnegative. For the lower bound, fix \(0{\lt}\varepsilon {\lt}1\) and choose \(\lambda _j{\gt}0\) with \(A_j\ge \lambda _jI\). Then \(A_j-(1-\varepsilon )c_jP=(1-\varepsilon )(A_j-c_jP)+\varepsilon A_j \ge \varepsilon \lambda _jI\), so a common \(\delta {\gt}0\) with \((1-\varepsilon )c_j\delta \le \varepsilon \lambda _j\) gives \(A_j\ge (1-\varepsilon )c_j(P+\delta I)\), a positive definite input. Monotonicity, homogeneity, the sum rule for the terminal weights, and the fact that a tree with a constant input \(X\) returns \(X\) give \(M\ge (1-\varepsilon )\bigl(\prod _jc_j^{w_j}\bigr)(P+\delta I) \ge (1-\varepsilon )\bigl(\prod _jc_j^{w_j}\bigr)P\); let \(\varepsilon \to 0\). For the band factorization, a matrix commuting with all inputs commutes with the root, so the roots of trees with inputs from different bands commute; induction over the tree with the product rule of Lemma 7.9.2 handles two bands, and induction on \(K\) handles the general case.
7.9.1 Logarithmic estimates for products and positive floors
Let \(A_0,\ldots ,A_{G-1}\) be positive definite operators on a finite-dimensional complex Hilbert space. Suppose that \(A_gA_h=A_hA_g\) whenever \(g\ne h\). Then
The empty product is the identity and the empty sum is zero; repeated operator values are allowed. This is the logarithmic identity used to separate commuting bands in [ Ope26b , Sections 7 and 8 ] .
Induct on \(G\). The empty case is \(\log I=0\). For the induction step, \(B=A_1\cdots A_{G-1}\) is positive definite and commutes with \(A_0\). The functional-calculus identity \(\log (A_0B)=\log A_0+\log B\) and the induction hypothesis give the result.
For every \(L\in \mathbb R\) and \(b\ge 0\),
If \(b{\gt}0\), then also
These estimates give explicit constant errors in the logarithmic floor comparison of [ Ope26b , Section 8 ] .
For the first bound, use \(b+e^L/2\ge e^L/2\) and take logarithms. For the second, both \(b\) and \(e^L\) are at most \(e^{\max \{ \log b,L\} }\), so \(b+e^L/2\le (3/2)e^{\max \{ \log b,L\} }\). Take logarithms, subtract \(\log b\), and use \(\max \{ \log b,L\} -\log b=\max \{ 0,L-\log b\} \).
Let \(T\) be a weighted binary tree with labels in a finite set \(J\) and terminal weights \(w_j\). Let \(b{\gt}0\), let \(U\in \mathbb R\), and suppose \(L_j\le U\) for every \(j\in J\). Set
Then
The constants are independent of the tree and its terminal probabilities. Zero weights and repeated leaf labels are allowed. These are the uniform scalar estimates preceding the logarithmic lower comparison in [ Ope26b , Section 8 ] .
Let \(T\) be a weighted binary tree with labels in a finite set \(J\) and terminal weights \(w_j\). Let \(b{\gt}0\), \(\kappa \ge 0\), and \(r,U\in \mathbb R\), and suppose \(L_j\le U\) for every \(j\in J\). For \(0\le \delta \le 1\), set
Then
The remainder \(R\) is independent of \(\delta \), the tree, and its terminal weights. This makes explicit the scalar error preceding the logarithmic lower comparison in [ Ope26b , Section 8 ] .
Apply Theorem 7.9.1.3 to the exponents \(\kappa L_j-r\), whose common upper bound is \(\kappa U-r\). Set \(E=|r|+|\log b|+\log (3/2)\ge 0\). Since the weights sum to one,
Multiply the second bound by \(\delta \ge 0\) and subtract it from the first. The inequalities \(\delta E\le E\) and \(r\le |r|\) give the stated remainder.
7.9.2 Normalized derivative maps
For \(0{\lt}p{\lt}1\) and a positive semidefinite matrix \(X\) the real power has the Löwner representation \(X^p=\kappa _p\int _0^\infty \bigl(t^{p-1}-t^p(t+X)^{-1}\bigr)\, dt\) with a positive constant \(\kappa _p\). A linear map on matrices is called completely positive when its entrywise action on every matrix amplification preserves positive semidefinite matrices.
Let \(0{\lt}p{\lt}1\) and let \(C\) be positive definite. Along Hermitian matrices, the map \(X\mapsto X^p\) is Fréchet differentiable at \(C\) with derivative
This derivative is completely positive and satisfies \(D(C^p)[C]=pC^p\).
Near \(C\) every shift \(t+X\) is invertible with \(\| (t+X)^{-1}\| \le 2(t+c)^{-1}\), where \(C\ge cI\); hence the derivative of the integrand is dominated by a multiple of \(t^p\bigl(t^{-1}-(t+c)^{-1}\bigr)\), which is integrable, and one may differentiate under the integral sign. Each integrand is a positive multiple of a sandwich map, so the integral is completely positive. The identity \(D(C^p)[C]=pC^p\) is the derivative at \(s=1\) of \((sC)^p=s^pC^p\).
Let \(A,B\) be positive definite, \(0\le p\le 1\) and \(M=A\# _pB\). At a child \(D\) of positive edge weight \(w\), the map \(\Phi _D(Z)=w^{-1}M^{-1/2}\, \partial _DM[D^{1/2}ZD^{1/2}]\, M^{-1/2}\) is unital and completely positive.
By the chain rule, \(\partial _BM[H]=A^{1/2}D(C^p)[A^{-1/2}HA^{-1/2}]A^{1/2}\) with \(C=A^{-1/2}BA^{-1/2}\), a composition of completely positive maps; at the endpoints \(p=0,1\) the derivative of the power is zero or the identity. Exchange symmetry gives \(\partial _A\). The identity \(D(C^p)[C]=pC^p\) gives \(\partial _BM[B]=pM\), and symmetrically \(\partial _AM[A]=(1-p)M\), which is unitality.
Let a weighted binary tree have distinct leaves with positive definite inputs \(A_j\), terminal weights \(w_j\) and root \(M\). The root is Fréchet differentiable in each input along Hermitian matrices. Composing the normalized vertex maps from a leaf \(j\) to the root gives a completely positive map \(\Phi _j\), the identity along an empty path, such that
and \(\Phi _j\) is unital when \(w_j{\gt}0\).
Since the leaves are distinct, at each vertex the input \(A_j\) enters only one child, and the chain rule applies with the partial derivatives of Lemma 7.9.2.2. The square-root normalizations cancel at successive vertices, so the normalized map at the root is the composition of the normalized edge maps along the path. The Euler identities propagate to \(\partial _{A_j}M[A_j]=w_jM\), which gives unitality.
If all inputs of a weighted binary tree with distinct leaves are block diagonal for a fixed orthogonal decomposition, the root is block diagonal and the normalized leaf maps preserve each matrix block. On the first diagonal block they are the normalized leaf maps of the tree of first diagonal blocks, and they do not depend on the other diagonal blocks.
The functional calculus of a block-diagonal Hermitian matrix acts blockwise, so the mean and the root are block diagonal. In the derivative integral every sandwich factor \((t+C)^{-1}\) is block diagonal, so the derivative of the power commutes with compression to each block and acts on the diagonal blocks as the derivatives at the diagonal blocks. These properties are preserved by sums, compositions, positive multiples and sandwiches by block diagonal matrices, which build the leaf maps.
For a leaf of positive terminal weight, the trace adjoint \(\Phi _j^\dagger \), defined by \(\operatorname {Tr}(\Phi _j^\dagger (\rho )Z)=\operatorname {Tr}(\rho \, \Phi _j(Z))\), maps density matrices to density matrices.
The trace adjoint of a positive map is positive, and \(\operatorname {Tr}\Phi _j^\dagger (\rho )=\operatorname {Tr}(\rho \, \Phi _j(I))=\operatorname {Tr}\rho \) by unitality.
7.9.3 Rectangular intertwiners of means and derivative maps
Throughout this subsection, the matrices act on finite-dimensional complex Hilbert spaces \(\mathcal H_1\) and \(\mathcal H_2\), and \(J:\mathcal H_2\to \mathcal H_1\) is an arbitrary linear map. Its adjoint is denoted by \(J^*\). No injectivity, surjectivity or isometry assumption is imposed. The identities concern the geometric means and derivative maps used in [ Ope26b , Section 7 ] .
Let \(L_1\) and \(L_2\) be Hermitian and satisfy \(L_1J=JL_2\). Then \(L_2J^*=J^*L_1\), and for every matrix \(H\) on \(\mathcal H_2\),
Taking adjoints gives the first identity. Substitute \(L_1J=JL_2\) and \(J^*L_1=L_2J^*\) into the left side of (9).
Let \(A\ge 0\) and \(B\ge 0\) satisfy \(AJ=JB\). For every real exponent \(s\), \(A^sJ=JB^s\). For negative exponents the spectral power is zero on the kernel.
The functional calculus of Hermitian matrices preserves rectangular intertwining. Apply it to the real spectral function \(x\mapsto x^s\).
Let \(A_1,B_1{\gt}0\) on \(\mathcal H_1\) and \(A_2,B_2{\gt}0\) on \(\mathcal H_2\) satisfy \(A_1J=JA_2\) and \(B_1J=JB_2\). For every real \(p\),
Let \(T\) be a fixed weighted binary tree whose vertex weights lie in \([0,1]\). If its positive definite inputs \(A_j,B_j\) satisfy \(A_jJ=JB_j\) for every label \(j\), and its roots are \(M_A=T(A)\) and \(M_B=T(B)\), then \(M_AJ=JM_B\). Repeated labels and zero vertex weights are permitted.
Intertwine the factors \(A^{1/2}\) and \(A^{-1/2}\) in the defining formula for the geometric mean, and then the power of \(A^{-1/2}BA^{-1/2}\). This proves (10); induction over the tree proves the root identity.
Let \(A{\gt}0\) and \(B{\gt}0\) satisfy \(AJ=JB\), and let \(0\le p\le 1\). Write \(\mathcal D_p(A)\) for the complex-linear extension of the derivative of the real power at \(A\). For arbitrary matrices \(H\) on \(\mathcal H_2\) and \(X\) on \(\mathcal H_1\),
The directions need not be Hermitian.
For \(0{\lt}p{\lt}1\), the shifted resolvents obey \((tI+A)^{-1}J=J(tI+B)^{-1}\) for \(t{\gt}0\). Apply Theorem 7.9.3.1 to their sandwiches and integrate the power-derivative formula. The endpoint maps are zero at \(p=0\) and the identity at \(p=1\). For compression, take adjoints of \(AJ=JB\) and apply the embedding identity to \(J^*\).
Suppose \(A_1,B_1{\gt}0\), \(A_2,B_2{\gt}0\), \(A_1J=JA_2\) and \(B_1J=JB_2\). Let \(0\le p\le 1\), and let \(\mathcal G_{p,L}(A,B)\) and \(\mathcal G_{p,R}(A,B)\) be the complex-linear extensions of the derivatives of \(A\# _pB\) in its first and second arguments. For each \(\nu \in \{ L,R\} \) and arbitrary matrices \(H\) and \(X\) on the respective spaces,
The same common intertwiner is used for both positive input pairs. The directions in the two arguments may be chosen independently.
For the second argument, intertwine the sandwiches by \(A^{1/2}\) and \(A^{-1/2}\) and the power derivative at the whitened input. The first-argument formula follows by exchanging the inputs and replacing \(p\) by \(1-p\). Apply the corresponding embedding identities to \(J^*\) to obtain compression.
Let \(T\) be a fixed weighted binary tree, and let \(A_i,B_i{\gt}0\) satisfy \(A_iJ=JB_i\) for every label \(i\). Denote its roots by \(M_A\) and \(M_B\), its recursive linear maps for a label \(j\) by \(\mathcal L_j^A\) and \(\mathcal L_j^B\). For either input family, the map at a leaf \(i\) is the identity when \(i=j\) and zero otherwise. At an internal vertex it is
where \(M_L,M_R\) are the child roots. If \(w_j\) is the total weight of label \(j\), summing over its occurrences, define
and define \(\Phi _j^B\) analogously. For arbitrary matrices \(H\) and \(X\) on the respective spaces,
Repeated labels are retained in the recursive sums. Zero total label weights are permitted, with reciprocal weight interpreted as zero; the normalized map of such a label is zero.
Induct over the tree, applying the two vertex derivative identities to the contributions of its children. Intertwine the normalizing sandwiches by \(A_j^{1/2},B_j^{1/2}\) and \(M_A^{-1/2},M_B^{-1/2}\) to obtain leaf-map covariance. Taking adjoints of each input intertwining equation gives the two compression identities.
Under the input hypotheses of Theorem 7.9.3.6, the trace adjoints obey, for every matrix \(\rho \) on \(\mathcal H_2\),
Repeated labels and zero total label weights are permitted. This is an embedding identity for arbitrary \(J\); it does not assert preservation of trace.
Applying \(\Phi _j^A\) and then compressing gives the same result as compressing first and applying \(\Phi _j^B\). Take trace adjoints, reversing the composition order. The trace adjoint of \(X\mapsto J^*XJ\) is \(\rho \mapsto J\rho J^*\).
7.9.4 The hyperbolic-sine Fourier multipliers
For \(0{\lt}s{\lt}1/2\) let \(g_s(z)=\sinh (sz)/\sinh (z/2)\) with \(g_s(0)=2s\), and \(m_s(u)=\sin (2\pi s)/(\cosh (2\pi u)+\cos (2\pi s))\).
For real \(z\), \(g_s(z)=\int _{\mathbb R}e^{iuz}m_s(u)\, du\), \(m_s(u){\gt}0\) and \(\int _{\mathbb R}m_s(u)\, du=2s\). For \(s=1/4\) the functions \(h_\pm (z)=(e^{\pm sz}-1)/(2\sinh (z/2))=\pm e^{\pm sz/2}g_{s/2}(z)\) have Fourier densities \(q_\pm (u)=\pm m_{s/2}(u\pm is/2)\) with \(|q_\pm (u)|\le m_s(u)/\sqrt2\). The quotients are interpreted continuously at zero.
For fixed real \(z\ne 0\), integrate \(e^{iuz}m_s(u)\) around the rectangle between \(\mathbb R\) and \(\mathbb R+i\). The poles in the strip are \(i(1/2-s)\) and \(i(1/2+s)\) with residues \(\pm 1/(2\pi i)\), and \(m_s(u+i)=m_s(u)\), so \((1-e^{-z})\int e^{iuz}m_s(u)\, du=e^{-(1/2-s)z}-e^{-(1/2+s)z}\). Shifting the contour of \(m_{s/2}\) by \(\pm is/2\) crosses no pole and gives \(q_\pm \): in the strip \(|\operatorname {Im}w|\le s/2\) one has \(|\cosh (2\pi w)+\cos (\pi s)|\ge \cos (\pi s)\cosh (2\pi \operatorname {Re}w)\). The shifted identities hold for every \(0{\lt}s{\lt}1/2\), in particular for \(s=1/4\).
If \(p\) lies strictly inside a rectangle, the boundary integral of \((u-p)^{-1}\) is \(2\pi i\). If \(f\) is holomorphic on the open rectangle away from finitely many interior points \(p\) and \(f-\sum _pc_p(u-p)^{-1}\) extends continuously to the closed rectangle, the boundary integral of \(f\) is \(2\pi i\sum _pc_p\).
On each edge a branch of the logarithm is a primitive of \((u-p)^{-1}\); the four edge contributions sum to the change of argument \(2\pi \). The regular part has vanishing boundary integral by the Cauchy–Goursat theorem.
The function \(g_s\) is continuous, \(m_s\) is positive and integrable with \(\int _{\mathbb R}m_s=2s\), for \(z\ne 0\) one has \((e^{\pm sz}-1)/(2\sinh (z/2))=\pm e^{\pm sz/2}g_{s/2}(z)\), and for \(s=1/4\), \(|m_{s/2}(u\pm is/2)|\le m_s(u)/\sqrt2\) for real \(u\).
The function \(u\mapsto \pi ^{-1}\arctan \bigl((e^{2\pi u}+\cos 2\pi s)/\sin 2\pi s\bigr)\) has derivative \(m_s\) and limits \(1/2-2s\) and \(1/2\) at \(-\infty \) and \(+\infty \). Near \(z=0\) both \(\sinh (sz)/z\) and \(\sinh (z/2)/z\) have limits \(s\) and \(1/2\). The factorization follows from \(e^{sz}-1=2e^{sz/2}\sinh (sz/2)\). At \(s=1/4\) and \(c=\cosh 2\pi u\), \(m_s(u)=c^{-1}\) and \(m_{s/2}(u\pm is/2)=(c+1\pm i\sinh 2\pi u)^{-1}\), whose modulus is \((2c(c+1))^{-1/2}\le (\sqrt2\, c)^{-1}\).
7.9.5 The filtered transport path
This subsection records the matrix part of the entropy and energy transport estimate of the area-law paper (Proposition 7.4, 06-transport.tex lines 377–434). Fix a mean tree \(T\) with leaves \(h\) and positive weights \(w_h\), a mean tree \(S_h\) for every \(h\), positive definite old inputs \(A_h\) and new inputs \(A'_{h,c}\). Let \(B_h\) be the root of \(S_h\) on the inputs \(A'_{h,c}\) and \(C_h=A_h^{-1/2}B_hA_h^{-1/2}\). For \(0\le p\le 1\) the interpolated tree replaces each leaf \(h\) of \(T\) by the vertex \(A_h\# _pB_h\); let \(M(p)\) be its root. For a nonzero vector \(x\) put \(N(p)=\| M(p)^{-1/4}x\| \), \(v(p)=M(p)^{-1/4}x/N(p)\), and define the transport states \(\sigma _{j,u}=\Phi _j^\dagger \bigl(|M^{-iu}v\rangle \langle M^{-iu}v|\bigr)\), where \(\Phi _j\) are the normalized derivative maps of the interpolated tree.
The transport states are density matrices. For \(0{\lt}p{\lt}1\),
The terminal weights of the interpolated tree are \(w_h(1-p)\) at the old leaf and \(w_hp\, \pi _c^{S_h}\) at the new leaves. Differentiating the vertex \(A_h\# _pB_h=A_h^{1/2}C_h^pA_h^{1/2}\) and the tree gives \(M^{-1/2}\partial _pMM^{-1/2}=\sum _hw_h\Phi _{(h,\mathrm{old})}(\log C_h)\) (06-transport.tex lines 442–468). In an eigenbasis of \(M\) the derivative of \(\| M^{-1/4}x\| ^2\) has coefficients \(g_{1/4}(\log x_r-\log x_t)\), and the Fourier formula for \(g_{1/4}\) turns them into the integral over \(u\) (lines 470–489). Passing each map to its trace adjoint gives the stated form (lines 490–494).
The function \(N\) is continuous and positive on \([0,1]\). For \(0\le p_0\le p_1\le 1\) the right side of the exact derivative is integrable on \([p_0,p_1]\), and its integral equals \(\log N(p_0)^2-\log N(p_1)^2\).
The derivative is bounded uniformly in \(p\) by \(\sum _hw_h\, \| \log C_h\| \int m_{1/4}\), since the transport states are density matrices. The fundamental theorem of calculus applies to the continuous function \(-\log N^2\) (06-transport.tex lines 769–779).
If \(Z\) commutes with every input of a mean tree and fixes the vector \(v\), then \(Z\sigma _{j,u}=\sigma _{j,u}\) for every leaf \(j\) and every real \(u\).
Every map in the construction of the normalized derivative maps is a right module map over \(Z\), so \(\Phi _j(XZ)=\Phi _j(X)Z\) and dually \(\Phi _j^\dagger (ZY)=Z\Phi _j^\dagger (Y)\). The matrix \(Z\) commutes with the root and hence fixes \(M^{-iu}v\).
7.9.6 Covariance of filtering and transport matrices
Let \(M{\gt}0\) and \(N{\gt}0\) satisfy \(MJ=JN\). For every real \(u\), \(M^{-iu}J=JN^{-iu}\). For a fixed weighted tree with positive inputs \(A_i,B_i\) satisfying \(A_iJ=JB_i\), roots \(M_A,M_B\), and any vector \(y\) in the corresponding Hilbert space, define, for \(q\in \{ A,B\} \),
The vector need not be normalized; repeated labels and zero total label weights are permitted. No isometry condition on \(J\) is required.
Functional calculus intertwines \(\log M\) and \(\log N\); the exponential of \(-iu\log M\) therefore intertwines with that of \(-iu\log N\). Apply this to the tree roots, then apply trace-adjoint covariance to the rank-one matrix of \(M_B^{-iu}v\).
Let \(M\ge 0\) and \(N\ge 0\) satisfy \(MJ=JN\). For every \(x\in \mathcal H_2\), the unnormalized filtered vector satisfies \(M^{-1/4}Jx=JN^{-1/4}x\). If in addition \(J^*J=I\), then
Negative powers vanish on the kernels. Each quotient in (25) is defined to be zero when its denominator is zero, so the identities include \(x=0\).
Apply real-power intertwining at exponent \(-1/4\). When \(J^*J=I\), the norm of every image \(Jy\) equals the norm of \(y\), so both normalization scalars coincide. Scalar multiplication commutes with \(J\), including the zero-scalar case.
Let a mean tree have positive weights \(\pi _j\), positive definite inputs \(A_j\) and root \(M\), let \(h\ge 0\) commute with \(A_j\) for every leaf \(j\) outside a finite set \(S\), and put \(W=\sum _{j\in S}\pi _j\). Let \(\mathsf D_j\) be the skew square of \(O_j=A_j^{-1/2}hA_j^{1/2}\). For every vector \(v\),
Factor \(h=T^*T\) with \(T=h^{1/2}\); the source factors through the last copy, and any factorization serves. An auxiliary second diagonal block \(B_j\) gives \(Y_j^*Z_j=O_j\) with \((Y_j-Z_j)^*(Y_j-Z_j)=\mathsf D_j\), and \(Y_j=Z_j\) off \(S\) (06-transport.tex lines 604–656). On the doubled space a block-rotation identity writes the rotated commutator as \(\sum _{j\in S}\pi _j\widehat\Phi _j(K_j)\). The Fourier densities \(q_\pm \), Cauchy–Schwarz and the Schwarz inequality for the unital two-positive leaf maps bound both error vectors by \(\tfrac 14W\sum _{j\in S}\pi _j\int m_{1/4}\operatorname {Tr}(\sigma _{j,u}\mathsf D_j)\) (lines 658–729), and the similarity pairing closes the estimate (lines 730–766).
7.10 Weighted means on complementary projection ranges
Let \(I\) be a finite set and let \(P\) be an orthogonal projection on \(\mathbb {C}^I\). The following statements give the two-range comparison used before taking logarithms in [ Ope26b , lines 603–621 of the norm-comparison argument ] .
For \(b,d{\gt}0\), the matrix \(b(\mathbb {1}-P)+dP\) is positive definite.
Both summands are positive semidefinite. Since \(P\) and \(\mathbb {1}-P\) are idempotent and \(P(\mathbb {1}-P)=(\mathbb {1}-P)P=0\), the matrix has inverse \(b^{-1}(\mathbb {1}-P)+d^{-1}P\). A positive semidefinite invertible matrix is positive definite.
Let \(T\) be a finite weighted binary tree with labels in a finite set \(J\) and terminal weights \(w_j\). For \(b{\gt}0\) and \(d_j{\gt}0\) for every \(j\in J\),
Repeated leaf labels and zero terminal weights are permitted.
Put \(B_j=b(\mathbb {1}-P)+d_jP\). By Theorem 7.10.1, each \(B_j\) is positive definite. The relations \(B_jP=P(d_j\mathbb {1})\) and \(B_j(\mathbb {1}-P)=(\mathbb {1}-P)(b\mathbb {1})\) pass through the tree by Theorem 7.9.3.3. Scalar homogeneity and evaluation of a constant family give
Adding the two identities proves (26).
With the same \(P\), \(b{\gt}0\), positive family \((d_j)_{j\in J}\), and tree \(T\), suppose matrices \(A_j\) satisfy \(b(\mathbb {1}-P)+d_jP\le A_j\) for every \(j\in J\). Then
The hypotheses impose no commutation condition on the matrices \(A_j\).
7.11 A logarithmic lower bound on complementary ranges
Let \(I\) and \(J\) be finite sets, let \(P\) be an orthogonal projection on \(\mathbb {C}^I\), and let \(T\) be a weighted binary tree with labels in \(J\) and terminal weights \(w_j\). Suppose \(b{\gt}0\), \(d_j{\gt}0\), and \(b(\mathbb {1}-P)+d_jP\le A_j\) for every \(j\in J\). Then
No commutation condition is imposed on the matrices \(A_j\). Repeated leaf labels and zero terminal weights are permitted. This is the operator comparison used before the projection-mass estimate in [ Ope26b , lines 603–640 of the norm-comparison argument ] .
Put \(r=\prod _{j\in J}d_j^{w_j}{\gt}0\). Theorem 7.10.3 gives \(b(\mathbb {1}-P)+rP\le T(A)\). The lower matrix is positive definite by Theorem 7.10.1, so monotonicity of the operator logarithm gives
Rewrite \(b(\mathbb {1}-P)+rP=b\mathbb {1}+(r-b)P\) and apply Theorem 7.19.1 with \(f=\log \). Its logarithm is \((\log b)(\mathbb {1}-P)+(\log r)P\). The identity \(\log r=\sum _{j\in J}w_j\log d_j\) proves (27).
7.12 Retained projection mass
Let \(P\) be an orthogonal projection on a finite-dimensional complex Hilbert space, and let \(T\) be a weighted binary tree with labels in a finite set \(J\) and terminal weights \(w_j\). Suppose \(b{\gt}0\) and, for each \(j\in J\), the real number \(d_j\) and matrix \(A_j\) satisfy
Let \(v\) be a unit vector and let \(\delta \in \mathbb R\) satisfy \(1-\delta \le \langle v,Pv\rangle \). Put \(\ell =\sum _{j\in J}w_j\log d_j\). Then
No upper bound on \(\delta \) or commutation of \(P\) with the inputs is required. This is the expectation step in the lower comparison of [ Ope26b , Section 8 ] .
The terminal-weight identities give \(\ell \ge \log b\). Set \(q=\langle v,Pv\rangle \). Taking the expectation in Theorem 7.11.1 gives
The last inequality uses \(q\ge 1-\delta \) and \(\ell -\log b\ge 0\).
7.13 Inverse powers from logarithmic expectations
These inequalities give the inverse-power step in the proof of the sharp comparison in [ Ope26b , lines 383–395 of the norm-comparison argument ] .
Let \(I\) be a finite set, let \(M\) be a positive definite matrix on \(\mathbb {C}^I\), and let \(\varphi \in \mathbb {C}^I\) satisfy \(\| \varphi \| =1\). For every \(s\in \mathbb {R}\),
The matrix \(M^{-2s}\) is positive definite, so its support is the whole space and \(\langle \varphi ,M^{-2s}\varphi \rangle {\gt}0\). Logarithmic Jensen, \(\log (M^{-2s})=-2s\log M\), and \(\langle \varphi ,M^{-2s}\varphi \rangle =\| M^{-s}\varphi \| ^2\) give
Exponentiation proves (28).
Let \(I\) be a finite set, let \(M\) be a positive definite matrix on \(\mathbb {C}^I\), and let \(p\in \mathbb {C}^I\) satisfy \(q=\| p\| ^2{\gt}0\). Put \(\varphi =p/\| p\| \). If \(L\in \mathbb {R}\) satisfies \(\operatorname {Re}\langle \varphi ,(\log M)\varphi \rangle \le L\), then, for every \(s\ge 0\),
The vector \(\varphi \) is a unit vector. Since \(-2s\le 0\), \(e^{-2sL}\le \exp (-2s\, \operatorname {Re}\langle \varphi ,(\log M)\varphi \rangle )\). Apply Theorem 7.13.1 and multiply by \(q\). Linearity gives \(q\| M^{-s}\varphi \| ^2=\| M^{-s}p\| ^2\).
7.14 The logarithm in the normalized inverse-power vector
The following inequalities give the final norm-to-logarithm step in the lower comparison of [ Ope26b , lines 642–651 of the norm-comparison argument ] .
Let \(I\) be a finite set, let \(M\) be a positive definite matrix on \(\mathbb {C}^I\), let \(s\in \mathbb {R}\), and let \(v\in \mathbb {C}^I\) be a unit vector. If \(N{\gt}0\) and \(N\| M^s v\| ^2\le 1\), then
Theorem 7.13.1, with exponent \(-s\), gives \(\exp (2s\, \operatorname {Re}\langle v,(\log M)v\rangle )\le \| M^s v\| ^2\). Multiplication by \(N\) yields a positive quantity bounded above by one. Taking logarithms proves the claim.
Let \(I\) be a finite set, let \(M\) be a positive definite matrix on \(\mathbb {C}^I\), let \(s\in \mathbb {R}\), and let \(p\in \mathbb {C}^I\) satisfy \(p\ne 0\) and \(\| p\| \le 1\). Put \(u=M^{-s}p\) and \(v=u/\| u\| \). Then
The opposite powers cancel: \(M^su=p\). Hence \(u\ne 0\) and \(\| v\| =1\). Linearity gives
Apply Theorem 7.14.1 with \(N=\| u\| ^2\).
7.15 Norm bounds from projection comparisons
Let \(J\) be finite and let \(g\) range over \(0,\ldots ,G-1\). For each \(g\), let \(T_g\) be a weighted binary tree with terminal weights \(w_{j,g}\) and positive definite inputs \(A_{j,g}\) on one finite-dimensional complex Hilbert space. Set \(M_g=T_g(A_{\cdot ,g})\) and \(M=M_0\cdots M_{G-1}\), and suppose the roots \(M_g\) commute pairwise. Let \(P_g\) be orthogonal projections and let \(b_g,c_{j,g}{\gt}0\) satisfy
For \(s\ge 0\) and a vector \(p\) with \(0{\lt}\| p\| \le 1\), put \(u=M^{-s}p\), \(v=u/\| u\| \), and \(\ell _g=\sum _{j\in J}w_{j,g}\log (b_g+c_{j,g}/2)\). If real numbers \(\delta _g\) satisfy \(1-\delta _g\le \langle v,P_gv\rangle \), then
The projections need not commute with the inputs, the roots, or one another. The empty product and zero terminal weights are included. This is an analytic implication of the floor, projection bounds, and mass estimates used in [ Ope26b , Section 8 ] .
Averaging the floor and projection bound gives \(b_g(\mathbb {1}-P_g)+(b_g+c_{j,g}/2)P_g\le A_{j,g}\). The roots and their commuting product are positive definite; \(M^sM^{-s}p=p\ne 0\) makes the normalization well defined. Theorem 7.12.1 bounds each \(\operatorname {Re}\langle v,(\log M_g)v\rangle \) from below. By Theorem 7.9.1.1, their sum is \(\operatorname {Re}\langle v,(\log M)v\rangle \). Multiply by \(2s\ge 0\) and apply Theorem 7.14.2.
Let \(J\) be finite and let \(g\) range over \(0,\ldots ,G-1\). Let \(T_g\) be weighted binary trees with terminal weights \(w_{j,g}\) and positive definite inputs \(A_{j,g}\) on one finite-dimensional complex Hilbert space. Suppose their roots \(M_g=T_g(A_{\cdot ,g})\) commute pairwise, set \(M=M_0\cdots M_{G-1}\), and let \(P_g\) be orthogonal projections. Let \(b_g{\gt}0\), \(\kappa _g\ge 0\), and let \(L_{j,g},U_g,r_g\in \mathbb R\) satisfy
For \(s\ge 0\) and \(0{\lt}\| p\| \le 1\), put \(u=M^{-s}p\) and \(v=u/\| u\| \). Suppose \(\delta _g\ge 0\) and \(1-\delta _g\le \langle v,P_gv\rangle \). Define \(R_g=2|r_g|+|\log b_g|+\log 2+\log (3/2)\). Then
The remainders are independent of the deficits and terminal weights; no upper bound on \(\delta _g\) is imposed. This gives explicit errors in the analytic lower comparison of [ Ope26b , Section 8 ] .
Set \(\varepsilon _g=\min \{ 1,\delta _g\} \). If \(\delta _g\le 1\), the mass bound is unchanged; otherwise positivity of \(P_g\) gives \(1-\varepsilon _g=0\le \langle v,P_gv\rangle \). Apply Theorem 7.15.1 with \(c_{j,g}=e^{\kappa _gL_{j,g}-r_g}\) and deficits \(\varepsilon _g\). Theorem 7.9.1.4 bounds each resulting summand. Finally, \(\varepsilon _g\le \delta _g\) and \(\kappa _g\max \{ 0,U_g\} \ge 0\) allow replacement of \(\varepsilon _g\) by \(\delta _g\) in the leading term.
7.16 Exponential bounds on invariant subspaces
The inverse metric comparison in [ Ope26b , Section 8 ] uses exponential inequalities on invariant subspaces. All matrices below act on finite-dimensional complex inner product spaces, which may have dimension zero. Matrix order means positive semidefinite order.
Let \(A,B\) be commuting Hermitian matrices. If \(A\leq B\), then \(e^A\leq e^B\).
Set \(D=B-A\geq 0\). Scalar functional calculus gives \(e^D\geq I\). Since \(A\) and \(D\) commute, \(e^B-e^A=e^A(e^D-I)\geq 0\).
Let \(A=A^*\) and let \(P=P^*=P^2\) commute with \(A\). Suppose that \(bP\leq PAP\) for some \(b\in \mathbb R\). If \(f\colon \mathbb R\to \mathbb R\) and \(c\in \mathbb R\) satisfy \(f(x)\leq c\) whenever \(x\geq b\), then \(Pf(A)P\leq cP\).
The Hermitian matrix \(B=PA+b(I-P)\) satisfies \(B\geq bI\). Functional calculus on its finite spectrum gives \(f(B)\leq cI\). Since \(PA=PB\), the restrictions of \(f(A)\) and \(f(B)\) to the range of \(P\) agree. Compressing the inequality gives the result.
Let \(M\) be a positive definite complex matrix on a finite-dimensional space, which may have dimension zero. Suppose that \(b{\gt}0\), \(s\le 0\), and \(bI\le M\). Then \(M^s\le b^sI\), with powers defined by spectral functional calculus.
For every \(x\ge b{\gt}0\), one has \(x^s\le b^s\). Apply Theorem 7.16.2 with \(P=I\) and \(f(x)=x^s\).
Under the hypotheses on \(A,P,b\) in Theorem 7.16.2, every \(a\geq 0\) satisfies \(Pe^{-aA}P\leq e^{-ab}P\).
Apply the theorem to \(f(x)=e^{-ax}\), which is at most \(e^{-ab}\) on \([b,\infty )\).
Let \(A\) and \(P\) be Hermitian, let \(B\) be a complex matrix, and assume \(AP=PA\) and \(AB=BA\). If \(Pe^BP\leq cP\) for \(c\in \mathbb R\), then every vector \(v\) with \(Pv=v\) satisfies \(\operatorname {Re}(v^*e^{A+B}v)\leq c\operatorname {Re}(v^*e^Av)\).
Set \(C=e^{A/2}\). Commutation gives \(P(Cv)=Cv\), and the fixed-vector quadratic bound applied to \(Cv\) gives \(\operatorname {Re}((Cv)^*e^B(Cv))\leq c\| Cv\| ^2\). Since \(C=C^*\), \(Ce^BC=e^{A+B}\) and \(C^2=e^A\), this is the desired inequality.
Under the matrix hypotheses of Theorem 7.16.5, \(Pe^BP\leq P\) implies \(\operatorname {Re}(v^*e^{A+B}v)\leq \operatorname {Re}(v^*e^Av)\) for every \(Pv=v\).
Take \(c=1\).
Let \(A,B\) be Hermitian, let \(P=P^*=P^2\), and suppose that \(AP=PA\) and \(AB=BA\). If \(Pe^BP\leq cP\) for \(c\in \mathbb R\), then \(Pe^{A+B}P\leq cPe^AP\).
Apply the fixed-vector comparison to \(Pv\) for every \(v\). The two compressed matrices are Hermitian, so their quadratic-form comparison is equivalent to their matrix order.
Under the hypotheses of Theorem 7.16.7, let \(\rho \geq 0\) satisfy \(P\rho =\rho \). Then \(\operatorname {Re}\operatorname {tr}(\rho e^{A+B}) \leq c\operatorname {Re}\operatorname {tr}(\rho e^A)\). No trace normalization is required.
Taking adjoints gives \(\rho P=\rho \). Pair the compressed matrix inequality with \(\rho \), using positivity of the trace pairing. Cyclicity and the two support identities remove both projections.
7.17 A lower bound from a compressed inverse
All operators in this section act on a finite-dimensional complex Hilbert space, which may have dimension zero. Operator inequalities refer to the positive-semidefinite order.
Let \(T\) be an operator and let \(c\ge 0\). If \(T^*T\le c\mathbb {1}\), then \(TT^*\le c\mathbb {1}\).
The hypothesis gives \(\| T\| ^2=\| T^*T\| \le c\). Thus \(TT^*\le \| T\| ^2\mathbb {1}\le c\mathbb {1}\).
Let \(A\) be positive definite, let \(P\) be an orthogonal projection, and let \(c{\gt}0\). Then
No commutation between \(A\) and \(P\) is required. This is the implication from the compressed-inverse estimate to the lower bound in [ Ope26b , Section 8 ] .
Put \(S=A^{1/2}\) and \(T=S^{-1}P\). Since \(0\le P\le \mathbb {1}\), the hypothesis gives \(T^*T=PA^{-1}P\le c\mathbb {1}\). Theorem 7.17.1 yields \(TT^*\le c\mathbb {1}\). Congruence by \(S\), together with \(ST=T^*S=P\), now gives
Dividing by \(c\) proves the assertion.
7.18 Sublinear remainders in the lower norm comparison
Fix real numbers \(s,G,t,c,C_f\) and a real sequence \((\varepsilon _k)\) such that \(\varepsilon _k/k\to 0\) as \(k\to \infty \). Put
Then \(E_k/k\to 0\). No sign condition on the five fixed coefficients or on \(\varepsilon _k\) is required.
This is the scalar limit for the explicit remainder in the rough lower norm estimate of [ Ope26b , Section 8 ] .
For every fixed real \(a\), the limit \(\log (k+a)/k\to 0\) follows by substituting \(x=k+a\) in \(\log x/(x-a)\to 0\). The cases \(a=1,2\), together with the assumption on \(\varepsilon _k\), give \(r_k/k\to 0\). Continuity of the absolute value gives \(|r_k|/k\to 0\). The remaining logarithmic and constant terms also vanish after division by \(k\); multiplication by the fixed coefficient \(2sG\) preserves the limit.
7.19 Functional calculus on complementary projection ranges
Let \(P\) be an orthogonal projection on a finite-dimensional complex Hilbert space. For arbitrary \(b,d\in \mathbb R\) and an arbitrary function \(f:\mathbb R\to \mathbb R\),
Here functional calculus uses the restriction of \(f\) to the finite spectrum. Neither continuity of \(f\) nor distinctness of \(b,d\) is required. Either projection range, or the entire space, may have dimension zero. This is the complementary-range calculation used in [ Ope26b , Section 7, lines 603–640 ] .
The projections \(I-P,P\) are orthogonal and sum to \(I\). Apply Lemma 23.4.1 to this resolution with scalar values \(b,d\). This argument permits zero projections and repeated scalar values. In particular, if \(b=d\), the formula reduces to \(f(b)I\); if \(P=0\) or \(P=I\), it reduces to \(f(b)I\) or \(f(d)I\), respectively.
7.20 Averaging matrix lower bounds
Let \(A,P\) be complex matrices on a finite-dimensional space and let \(b,c\in \mathbb R\). If \(2bI\le A\) and \(cP\le A\), then
No projection or commutation assumption on \(P\) is needed. This is the averaging step preceding the logarithmic comparison in [ Ope26b , Section 7, lines 603–615 ] .
Add the two assumed inequalities and divide by two. The resulting lower matrix \(bI+(c/2)P\) equals the displayed expression.
Let \(T,S\) be complex matrices on a finite-dimensional space and let \(c\in \mathbb R\). If \(S\le I\), \(c\le 1\) and \(cS\le T\), then
In particular, this applies to extension by the identity off the range of an orthogonal projection. The inequality itself does not require \(S\) to be a projection. This auxiliary order inequality extends a lower bound to the whole space.
Since \(I-S\ge 0\) and \(c\le 1\), one has \(c(I-S)\le I-S\). Add this inequality to \(cS\le T\).