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Chapter
Some Applications
Matrices are important in many areas of both pure and applied mathematics. In particular, they play essential roles in quantum probability and quantum information.
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Chapter
Map**s and Algebras
Most of the statements and definitions in this chapter are formulated in the Hilbert space setting.
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Chapter
Matrix Monotone Functions and Convexity
Let \((a, b) \subset {\mathbb R}\) ( a , ...
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Chapter
Majorization and Singular Values
A citation from von Neumann: “The object of this note is the study of certain properties of complex matrices of \(n\) ...
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Article
Operator log-convex functions and operator means
We study operator log-convex functions on (0, ∞), and prove that a continuous nonnegative function on (0, ∞) is operator log-convex if and only if it is operator monotone decreasing. Several equivalent conditi...
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Article
Free Analog of Pressure and Its Legendre Transform
The free analog of the pressure is introduced for multivariate noncommutative random variables and its Legendre transform is compared with Voiculescu’s microstate free entropy.
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Article
Free transportation cost inequalities via random matrix approximation
By means of random matrix approximation procedure, we re-prove Biane and Voiculescu’s free analog of Talagrand’s transportation cost inequality for measures on R in a more general setup. Furthermore, we prove the...
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Book
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Chapter
A Appendices
A.1 Non-symmetric means
A.2 Norm inequality for operator integrals
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Chapter
References
Abstract not available
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Chapter
2 Double integral transformations
2.1 Schur multipliers and Peller’s theorem
2.2 Extension to B(H)
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Chapter
4 Convergence of means
4.1 Main convergence result
4.2 Related convergence results
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Chapter
6 Heinz-type means A α
6.1 Norm continuity in parameter
6.2 Convergence of operator Riemann sums
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Chapter
8 Certain alternating sums of operators
8.1 Preliminaries
8.2 Uniform bounds for norms
8.3 Mono...
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Chapter
1 Introduction
The present monograph is devoted to a thorough study of means for Hilbert space operators, especially comparison of (unitarily invariant) norms of operator means and their convergence properties in various asp...
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Chapter
3 Means of operators and their comparison
3.1 Symmetric homogeneous means
3.2 Integral expression and comparison of norms
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Chapter
5 A-L-G interpolation means M α
5.1 Monotonicity and related results
5.2 Characterization of |||M ∞(H,K)X<∞
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Chapter
7 Binomial means B α
7.1 Majorization B α⪯M ∞
7.2 Equivalence of |||B α (H,K)X||| for
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Article
Properties of Free Entropy Related to Polar Decomposition
The free entropies of non-selfadjoint random variables and of unitary random variables are introduced and discussed by the methods of V...
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Article
Trace norm convergence of exponential product formula
LetH andK be lower-bounded self-adjoint operators whose form sum is denoted byH \(\hat + \) K. We show the norm in...
