Operator Lipschitz functions on Banach spaces

Let $X$, $Y$ be Banach spaces and let $\mathcal{L}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. We develop the theory of double operator integrals on $\mathcal{L}(X,Y)$ and apply this theory to obtain commutator estimates of the form $\|f(B)S-Sf(A)\|_{\mathcal{L}(X,Y)}\leq \textrm{const} \|BS-SA\|_{\mathcal{L}(X,Y)}$ for a large class of functions $f$, where $A\in\mathcal{L}(X)$, $B\in \mathcal{L}(Y)$ are scalar type operators and $S\in \mathcal{L}(X,Y)$. In particular, we establish this estimate for $f(t):=|t|$ and for diagonalizable operators on $X=\ell_{p}$ and $Y=\ell_{q}$, for $p<q$ and $p=q=1$, and for $X=Y=\mathrm{c}_{0}$. We also obtain results for $p\geq q$. We also study the estimate above in the setting of Banach ideals in $\mathcal{L}(X,Y)$. The commutator estimates we derive hold for diagonalizable matrices with a constant independent of the size of the matrix.Comment: Final version published in Studia Mathematica, with some minor change

Peller's problem concerning Koplienko-Neidhardt trace formulae: the unitary case

We prove the existence of a complex valued $C^2$-function on the unit circle, a unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class $S^2$, such that the perturbated operator $$f(e^{iZ}U)-f(U) -\frac{d}{dt}\bigl(f(e^{itZ}U)\bigr)_{\vert t=0}$$ does not belong to the space $S^1$ of trace class operators. This resolves a problem of Peller concerning the validity of the Koplienko-Neidhardt trace formula for unitaries

Resolution of Peller's problem concerning Koplienko-Neidhardt trace formulae

A formula for the norm of a bilinear Schur multiplier acting from the Cartesian product $\mathcal S^2\times \mathcal S^2$ of two copies of the Hilbert-Schmidt classes into the trace class $\mathcal S^1$ is established in terms of linear Schur multipliers acting on the space $\mathcal S^\infty$ of all compact operators. Using this formula, we resolve Peller's problem on Koplienko-Neidhardt trace formulae. Namely, we prove that there exist a twice continuously differentiable function $f$ with a bounded second derivative, a self-adjoint (unbounded) operator $A$ and a self-adjoint operator $B\in \mathcal S^2$ such that $$ f(A+B)-f(A)-\frac{d}{dt}(f(A+tB))\big\vert_{t=0}\notin \mathcal S^1. $

Multilinear operator integrals: theory and applications

This book provides a comprehensive treatment of multilinear operator integral techniques. The exposition is structured to be suitable for a course on methods and applications of multilinear operator integrals and also as a research aid. The ideas and contributions to the field are surveyed and up-to-date results and methods are presented. Most practical constructions of multiple operator integrals are included along with fundamental technical results and major applications to smoothness properties of operator functions (Lipschitz and Hölder continuity, differentiability), approximation of operator functions, spectral shift functions, spectral flow in the setting of noncommutative geometry, quantum differentiability, and differentiability of noncommutative L^p-norms. Main ideas are demonstrated in simpler cases, while more involved, technical proofs are outlined and supplemented with references. Selected open problems in the field are also presented

Multiple operator integrals: development and applications

Double operator integrals, originally introduced by Y.L. Daletskii and S.G. Krein in 1956, have become an indispensable tool in perturbation and scattering theory. Such an operator integral is a special mapping defined on the space of all bounded linear operators on a Hilbert space or, when it makes sense, on some operator ideal. Throughout the last 60 years the double and multiple operator integration theory has been greatly expanded in different directions and several definitions of operator integrals have been introduced reflecting the nature of a particular problem under investigation.The present thesis develops multiple operator integration theory and demonstrates how this theory applies to solving of severaldeep problems in Noncommutative Analysis.The first part of the thesis considers double operator integrals. Here we present the key definitions and prove several importantproperties of this mapping. In addition, we give a solution of the Arazy conjecture, which was made by J. Arazy in 1982. In this partwe also discuss the theory in the setting of Banach spaces and, as an application, we study the operator Lipschitz estimateproblem in the space of all bounded linear operators on classical Lp-spaces of scalar sequences. The second part of the thesis develops important aspects of multiple operator integration theory. Here, we demonstrate how this theory applies to a solution of the problem on a Koplienko-Neidhardt trace formulae for a Taylor remainder of order two, which wasraised by V. Peller in 2005, and also extend the solution for a Taylor remainder of an arbitrary order. Finally, using the tools from multiple operator integration theory, we present an affirmative solution of a question concerning Frechet differentiability of the norm of Lp-spaces, which has been of interest to experts in Banach space geometry for the last 50 years. We resolve this question in the most general setting, namely for the non-commutative Lp-spaces associated with an arbitrary von Neumann algebra, thus answering the open question suggested by G. Pisier and Q. Xu in their influential survey on the geometry of such spaces