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Register a new userGeneralizing Liebs Concavity Theorem via Operator Interpolation
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De Huang
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We introduce the notion of $k$-trace and use interpolation of operators to prove the joint concavity of the function $(A,B)mapstotext{Tr}_kbig[(B^frac{qs}{2}K^*A^{ps}KB^frac{qs}{2})^{frac{1}{s}}big]^frac{1}{k}$, which generalizes Liebs concavity theorem from trace to a class of homogeneous functions $text{Tr}_k[cdot]^frac{1}{k}$. Here $text{Tr}_k[A]$ denotes the $k_{text{th}}$ elementary symmetric polynomial of the eigenvalues of $A$. This result gives an alternative proof for the concavity of $Amapstotext{Tr}_kbig[exp(H+log A)big]^frac{1}{k}$ that was obtained and used in a recent work to derive expectation estimates and tail bounds on partial spectral sums of random matrices.
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We show that Liebs concavity theorem holds more generally for any unitarily invariant matrix function $phi:mathbf{H}^n_+rightarrow mathbb{R}$ that is monotone and concave. Concretely, we prove the joint concavity of the function $(A,B) mapstophibig[(B^frac{qs}{2}K^*A^{ps}KB^frac{qs}{2})^{frac{1}{s}}big] $ on $mathbf{H}_+^mtimesmathbf{H}_+^n$, for any $Kin mathbb{C}^{mtimes n},sin(0,1],p,qin[0,1], p+qleq 1$.
We show that Liebs concavity theorem holds more generally for any unitary invariant matrix function $phi:mathbf{H}_+^nrightarrow mathbb{R}_+^n$ that is concave and satisfies Holders inequality. Concretely, we prove the joint concavity of the function $(A,B) mapstophibig[(B^frac{qs}{2}K^*A^{ps}KB^frac{qs}{2})^{frac{1}{s}}big] $ on $mathbf{H}_+^ntimesmathbf{H}_+^m$, for any $Kin mathbb{C}^{ntimes m}$ and any $s,p,qin(0,1], p+qleq 1$. This result improves a recent work by Huang for a more specific class of $phi$.
We prove a Caratheodory-Fejer type interpolation theorem for certain matrix convex sets in $C^d$ using the Blecher-Ruan-Sinclair characterization of abstract operator algebras. Our results generalize the work of Dmitry S. Kalyuzhnyi-Verbovetzkii for the d-dimensional non-commutative polydisc.
We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends to higher multiplicity (e.g., multiframes) their dilation approach. We prove several results for operator-valued frames concerning their parametrization, duality, disjointeness, complementarity, and composition and the relationship between the two types of similarity (left and right) of such frames. We then apply these notions to prove that the collection of multiframe generators for the action of a discrete group on a Hilbert space is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. The proof is obtained by parametrizing this collection by a class of partial isometries in a larger von Neumann algebra. In the multiplicity one case this class reduces to the unitary class which is path-connected in norm, but in the infinite multiplicity case this class is path connected only in the strong operator topology and the proof depends on properties of tensor product slice maps.
Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In this paper we deeply study such behavior not only for operator monotone functions but also for operator convex functions on bounded intervals. More precisely, we prove that if $f$ is a nonlinear operator convex function on a bounded interval $(a,b)$ and $A, B$ are bounded linear operators acting on a Hilbert space with spectra in $(a,b)$ and $A-B$ is invertible, then $sf(A)+(1-s)f(B)>f(sA+(1-s)B)$. A short proof for a similar known result concerning a nonconstant operator monotone function on $[0,infty)$ is presented. Another purpose is to find a lower bound for $f(A)-f(B)$, where $f$ is a nonconstant operator monotone function, by using a key lemma. We also give an estimation of the Furuta inequality, which is an excellent extension of the Lowner--Heinz inequality.
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