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Aleksei Aleksandrov

Publications and source records attributed to Aleksei Aleksandrov.

At least 19 recordsLinked to original sources
arXiv2025

Analytic Schur multipliers

We study in this paper analytic Schur multipliers on ${\Bbb C}_+^2$ and ${\Bbb D}^2$, i.e. Schur multipliers on ${\Bbb R}^2$ and ${\Bbb T}^2$ that are boundary-value functions of functions analytic in ${\Bbb C}_+^2$ and ${\Bbb D}^2$. Such Schur multipliers are important when studying properties of functions of maximal dissipative operators and contractions under perturbation. We show that if the boundary-value function of a Schur multiplier has certain regularity properties, then it can be represented as an element of the Haagerup tensor product of spaces of analytic functions with similar regularity properties.

math.FA↗
arXiv2025

Functions of dissipative operators under relatively bounded and relatively trace class perturbations

We study the behaviour of functions of dissipative operators under relatively bounded and relatively trace class perturbation. We introduce and study the class of analytic relatively operator Lipschitz functions. An essential role is played by double operator integrals with respect to semispectral measures. We also study the class of analytic resolvent Lipschitz functions. Then we obtain a trace formula in the case of relatively trace class perturbations and show that the maximal class of function for which the trace formula holds in the case of relatively trace class perturbations coincides with the class of analytic relatively operator Lipschitz functions. We also establish the inequality $\int|\boldsymbolξ(t)|(1+|t|)^{-1}\,{\rm d}t<\infty$ for the spectral shift function $\boldsymbolξ$ in the case of relatively trace class perturbations.

math.FA↗
arXiv2025

Functions of self-adjoint operators under relatively bounded and relatively trace class perturbations. Relatively operator Lipschitz functions

We study the behaviour of functions of self-adjoint operators under relatively bounded and relatively trace class perturbation We introduce and study the class of relatively operator Lipschitz functions. An essential role is played by double operator integrals. We also consider study the class of resolvent Lipschitz functions. Then we obtain a trace formula in the case of relatively trace class perturbations and show that the maximal class of function for which the trace formula holds in the case of relatively trace class perturbations coincides with the class of relatively operator Lipschitz functions. Our methods also gives us a new approach to the inequality $\int|\boldsymbolξ(t)|(1+|t|)^{-1}\,{\rm d}t<\infty$ for the spectral shift function $\boldsymbolξ$ in the case of relatively trace class perturbations.

math.FA↗
arXiv2022

Triangular projection on $\boldsymbol{S}_p,~0<p<1,$ and related inequalities

In this paper we study properties of the triangular projection ${\mathcal P}_n$ on the space of $n\times n$ matrices. The projection ${\mathcal P}_n$ annihilates the entries of an $n\times n$ matrix below the main diagonal and leaves the remaining entries unchanged. We estimate the $p$-norms of ${\mathcal P}_n$ as an operator on the Schatten--von Neumann class $\boldsymbol{S}_p$ for $0<p<1$. The main result of the paper shows that for $p\in(0,1)$, the $p$-norms of ${\mathcal P}_n$ on $\boldsymbol{S}_p$ behave as $n\to\infty$ as $n^{1/p-1}$. This solves a problem posed by B.S. Kashin. Among other results of this paper we mention the result that describes the behaviour of the $\boldsymbol{S}_p$-quasinorms of the $n\times n$ matrices whose entries above the diagonal are equal to 1 while the entries below the diagonal are equal to 0.

math.FA↗
arXiv2022

Functions of pairs of unbounded noncommuting self-adjoint operators under perturbation

For a pair $(A,B)$ of not necessarily bounded and not necessarily commuting self-adjoint operators and for a function $f$ on the Euclidean space ${\Bbb R}^2$ that belongs to the inhomogeneous Besov class $B_{\infty,1}^1({\Bbb R}^2)$, we define the function $f(A,B)$ of these operators as a densely defined operator. We consider the problem of estimating the functions $f(A,B)$ under perturbations of the pair $(A,B)$. It is established that if $1\le p\le2$, and $(A_1,B_1)$ and $(A_2,B_2)$ are pairs of not necessarily bounded and not necessarily commuting self-adjoint operators such that the operators $A_1-A_2$ and $B_1-B_2$ belong to the Schatten--von Neumann class $\boldsymbol{S}_p$ with $p\in[1,2]$ and $f\in B_{\infty,1}^1({\Bbb R}^2)$, then the following Lipschitz type estimate holds: \[ \|f(A_1,B_1)-f(A_2,B_2)\|_{\boldsymbol{S}_p} \le\operatorname{const}\|f\|_{B_{\infty,1}^1}\max\big\{\|A_1-A_2\|_{\boldsymbol{S}_p},\|B_1-B_2\|_{\boldsymbol{S}_p}\big\}. \]

math.FA↗
arXiv2022

Functons of perturbed pairs of dissipative operators

Let $f$ be a function in the inhomogeneous analytic Besov space $B_{\infty,1}^1$. For a pair $(L,M)$ of not necessarily commuting maximal dissipative operators, we define the function $f(L,M)$ of $L$ and $M$ as a densely defined linear operator. We prove for $p\in[1,2]$ that if $(L_1,M_1)$ and $(L_2,M_2)$ are pairs of not necessarily commuting maximal dissipative operators such that both differences $L_1-L_2$ and $M_1-M_2$ belong to the Schatten--von Neumann class $\boldsymbol{S}_p$ than for an arbitrary function $f$ in the inhomogeneous analytic Besov space $B_{\infty,1}^1$, the operator difference $f(L_1,M_1)-f(L_2,M_2)$ belongs to $\boldsymbol{S}_p$ and the following Lipschitz type estimate holds: $$ \|f(L_1,M_1)-f(L_2,M_2)\|_{\boldsymbol{S}_p} \le\operatorname{const}\|f\|_{B_{\infty,1}^1}\max\big\{\|L_1-L_2\|_{\boldsymbol{S}_p},\|M_1-M_2\|_{\boldsymbol{S}_p}\big\}. $$

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arXiv2021

Functions of perturbed noncommuting unbounded self-adjoint operators

Let $f$ be a function on ${\Bbb R}^2$ in the inhomogeneous Besov space $B_{\infty,1}^1({\Bbb R}^2)$. For a pair $(A,B)$ of not necessarily bounded and not necessarily commuting self-adjoint operators, we define the function $f(A,B)$ of $A$ and $B$ as a densely defined linear operator. We show that if $1\le p\le2$, $(A_1,B_1)$ and $(A_2,B_2)$ are pairs of not necessarily bounded and not necessarily commuting self-adjoint operators such that both $A_1-A_2$ and $B_1-B_2$ belong to the Schatten--von Neumann class $\boldsymbol{S}_p$ and $f$ is in the above inhomogeneous Besov space, then the following Lipschitz type estimate holds: $$ \|f(A_1,B_1)-f(A_2,B_2)\|_{\boldsymbol{S}_p} \le\operatorname{const}\max\big\{\|A_1-A_2\|_{\boldsymbol{S}_p},\|B_1-B_2\|_{\boldsymbol{S}_p}\big\}. $$

math.FA↗
arXiv2019

Schur multipliers of Schatten--von Neumann classes $\boldsymbol{S_p}$

We study in this paper properties of Schur multipliers of Schatten von Neumann classes $\boldsymbol{S}_p$. We prove that for $p\le1$, Schur multipliers of $\boldsymbol{S}_p$ are necessarily completely bounded. We also introduce for $p\le1$ a scale ${\mathscr W}_p$ of tensor products of $\ell^\infty$ and prove that matrices in ${\mathscr W}_p$ are Schur multipliers of $\boldsymbol{S}_p$. We compare this sufficient condition with the sufficient condition of membership in the $p$-tensor product of $\ell^\infty$ spaces.

math.FA↗
arXiv2019

Functions of noncommuting operators under perturbation of class $\boldsymbol{S}_p$

In this article we prove that for $p>2$, there exist pairs of self-adjoint operators $(A_1,B_1)$ and $(A_2,B_2)$ and a function $f$ on the real line in the homogeneous Besov class $B_{\infty,1}^1({\Bbb R}^2)$ such that the differences $A_2-A_1$ and $B_2-B_1$ belong to the Schatten--von Neumann class $\boldsymbol{S}_p$ but $f(A_2,B_2)-f(A_1,B_1)\not\in\boldsymbol{S}_p$. A similar result holds for functions of contractions. We also obtain an analog of this result in the case of triples of self-adjoint operators for any $p\ge1$

math.FA↗
arXiv2018

Functions of perturbed pairs of noncommuting contractions

We consider functions $f(T,R)$ of pairs of noncommuting contractions on Hilbert space and study the problem for which functions $f$ we have Lipschitz type estimates in Schatten--von Neumann norms. We prove that if $f$ belongs to the Besov class $(B_{\infty,1}^1)_+({\Bbb D}^2)$ of analytic functions in the bidisk, then we have a Lipschitz type estimate for functions $f(T,R)$ of pairs of not necessarily commuting contractions $(T,R)$ in the Schatten--von Neumann norms $\boldsymbol{S}_p$ for $p\in[1,2]$. On the other hand, we show that for functions in the Besov space $(B_{\infty,1}^1)_+({\Bbb D}^2)$, there are no Lipschitz such type estimates for $p>2$ as well as in the operator norm.

math.FA↗
arXiv2018

Dissipative operators and operator Lipschitz functions

The purpose of this paper is to obtain an integral representation for the difference $f(L_1)-f(L_2)$ of functions of maximal dissipative operators. This representation in terms of double operator integrals will allow us to establish Lipschitz type estimates for functions of maximal dissipative operators. We also consider a similar problem for quasicommutators, i.e., operators of the form $f(L_1)R-Rf(L_2)$.

math.FA↗
arXiv2016

Operator Lipschitz Functions

The purpose of this survey article is a comprehensive study of operator Lipschitz functions. A continuous function $f$ on the real line ${\Bbb R}$ is called operator Lipschitz if $\|f(A)-f(B)\|\le{\rm const}\|A-B\|$ for arbitrary self-adjoint operators $A$ and $B$. We give sufficient conditions and necessary conditions for operator Lipschitzness. We also study the class of operator differentiable functions on ${\Bbb R}$. Then we consider operator Lipschitz functions on closed subsets of the plane as well as commutator Lipschitz functions on such subsets. Am important role is played by double operator integrals and Schur multipliers.

math.FA↗
arXiv2015

Triple operator integrals in Schatten--von Neumann norms and functions of perturbed noncommuting operators

We study perturbations of functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the following Lipschitz type estimate in the Schatten--von Neumann norm $\bS_p$, $1\le p\le2$ norm: $\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_p}\le\const(\|A_1-A_2\|_{\bS_p}+\|B_1-B_2\|_{\bS_p})$. However, the condition $f\in B_{\be,1}^1(\R^2)$ does not imply the Lipschitz type estimate in $\bS_p$ with $p>2$. The main tool is Schatten--von Neumann norm estimates for triple operator integrals.

math.FA↗
arXiv2014

Almost commuting functions of almost commuting self-adjoint operators

Let $A$ and $B$ be almost commuting (i.e, $AB-BA\in\bS_1$) self-adjoint operators. We construct a functional calculus $\f\mapsto\f(A,B)$ for $\f$ in the Besov class $B_{\be,1}^1(\R^2)$. This functional calculus is linear, the operators $\f(A,B)$ and $ψ(A,B)$ almost commute for $\f,\,ψ\in B_{\be,1}^1(\R^2)$, $\f(A,B)=u(A)v(B)$ whenever $\f(s,t)=u(s)v(t)$, and the Helton--Howe trace formula holds. The main tool is triple operator integrals.

math.FA↗
arXiv2014

Functions of perturbed noncommuting self-adjoint operators

We consider functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the following Lipschitz type estimate in the trace norm: $\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_1}\le\const(\|A_1-A_2\|_{\bS_1}+\|B_1-B_2\|_{\bS_1})$. However, the condition $f\in B_{\be,1}^1(\R^2)$ does not imply the Lipschitz type estimate in the operator norm.

math.FA↗
arXiv2011

Operator and commutator moduli of continuity for normal operators

We study in this paper properties of functions of perturbed normal operators and develop earlier results obtained in \cite{APPS2}. We study operator Lipschitz and commutator Lipschitz functions on closed subsets of the plane. For such functions we introduce the notions of the operator modulus of continuity and of various commutator moduli of continuity. Our estimates lead to estimates of the norms of quasicommutators $f(N_1)R-Rf(N_2)$ in terms of $\|N_1R- RN_2\|$, where $N_1$ and $N_2$ are normal operator and $R$ is a bounded linear operator. In particular, we show that if $0<\a<1$ and $f$ is a Hölder function of order $\a$, then for normal operators $N_1$ and $N_2$, $$ \|f(N_1)R-Rf(N_2)\|\le\const(1-\a)^{-2}\|f\|_{Ł_\a}\|N_1R-RN_2\|^\a\|R\|^{1-\a}. $$ In the last section we obtain lower estimates for constants in operator Hölder estimates.

math.FA↗
arXiv2011

Estimates of operator moduli of continuity

In \cite{AP2} we obtained general estimates of the operator moduli of continuity of functions on the real line. In this paper we improve the estimates obtained in \cite{AP2} for certain special classes of functions. In particular, we improve estimates of Kato \cite{Ka} and show that $$ \big\|\,|S|-|T|\,\big\|\le C\|S-T\|\log(2+\log\frac{\|S\|+\|T\|}{\|S-T\|}) $$ for every bounded operators $S$ and $T$ on Hilbert space. Here $|S|\df(S^*S)^{1/2}$. Moreover, we show that this inequality is sharp. We prove in this paper that if $f$ is a nondecreasing continuous function on $\R$ that vanishes on $(-\be,0]$ and is concave on $[0,\be)$, then its operator modulus of continuity $Ø_f$ admits the estimate $$ Ø_f(\d)\le\const\int_e^\be\frac{f(\d t)\,dt}{t^2\log t},\quad\d>0. $$ We also study the problem of sharpness of estimates obtained in \cite{AP2} and \cite{AP4}. We construct a $C^\be$ function $f$ on $\R$ such that $\|f\|_{L^\be}\le1$, $\|f\|_{\Li}\le1$, and $$ Ø_f(\d)\ge\const\,\d\sqrt{\log\frac2\d},\quad\d\in(0,1]. $$ In the last section of the paper we obtain sharp estimates of $\|f(A)-f(B)\|$ in the case when the spectrum of $A$ has $n$ points. Moreover, we obtain a more general result in terms of the $\e$-entropy of the spectrum that also improves the estimate of the operator moduli of continuity of Lipschitz functions on finite intervals, which was obtained in \cite{AP2}.

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