arXiv subjects

Anna Tomskova

Publications and source records attributed to Anna Tomskova.

5 recordsLinked to original sources

arXiv2025

Fuglede theorem for symmetric spaces of $τ$-measurable operators

We extend the classical Fuglede commutativity theorem to the full scale of symmetrically normed operator ideals. Our main result provides a complete characterization: a symmetric ideal or symmetric operator space of $τ$-measurable operators satisfies the Fuglede theorem if and only if its commutative core has non-trivial Boyd indices, or equivalently, if it is an interpolation space in the scale of $L_p$-spaces for $1<p<\infty$. This criterion subsumes all previously known cases, including Lorentz and Schatten classes.

math.FA↗

arXiv2016

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.

math.FA↗

arXiv2015

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.

math.FA↗

arXiv2015

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 $$ f(A+B)-f(A)-\frac{d}{dt}(f(A+tB))\big\vert_{t=0}\notin \mathcal S^1. $$

math.FA↗

arXiv2012

(E,F)-multipliers and applications

For two given symmetric sequence spaces $E$ and $F$ we study the $(E,F)$-multiplier space, that is the space all of matrices $M$ for which the Schur product $M\ast A$ maps $E$ into $F$ boundedly whenever $A$ does. We obtain several results asserting continuous embedding of $(E,F)$-multiplier space into the classical $(p,q)$-multiplier space (that is when $E=l_p$, $F=l_q$). Furthermore, we present many examples of symmetric sequence spaces $E$ and $F$ whose projective and injective tensor products are not isomorphic to any subspace of a Banach space with an unconditional basis, extending classical results of S. Kwapień and A. Pełczyński and of G. Bennett for the case when $E=l_p$, $F=l_q$.

math.FA↗