Related papers: Estimates on singular values of functions of pertu…

Spectral properties of bounded linear operators play a crucial role in several areas of mathematics and physics. For each self-adjoint, trace-class operator $O$ we define a set $\Lambda_n\subset \mathbb{R}$, and we show that it converges to…

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…

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…

We show that for a given pure contraction $T_7$ acting on a Hilbert space $\mathcal{H}$, if $(\tilde{F}_1, \dots, \tilde{F}_6) \in \mathcal{B}(\mathcal{D}_{T^*_7})$ with $[\tilde{F}_i, \tilde{F}_j] = 0, [\tilde{F}^*_i, \tilde{F}_{7-j}] =…

For an open set $\Om \subset \R^2$ let $\lambda(\Om)$ denote the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Om)$. Let $w_\Om$ be the torsion function for $\Om$, and let $\|.\|_p$ denote the $L^p$ norm. It is shown…

Let $A(x): =(A_{i, j}(x))$ be a continuous function defined on some subshift of $\Omega:= \{0,1, \cdots, m-1\}^\mathbb{N}$, taking $d\times d$ non-negative matrices as values and let $\nu$ be an ergodic $\sigma$-invariant measure on the…

The function $t \mapsto E_{\alpha}(\lambda t^\alpha)$ is widely regarded as the fractional analogue of the exponential function, yet its algebraic properties remain poorly understood. In particular, standard references lack a rigorous proof…

In [24, Theorem 2'] Charles Fefferman has proved the weak (1,1) boundedness for a class of oscillating singular integrals that includes the oscillating spectral multipliers of the Euclidean Laplacian $\Delta,$ namely, operators of the form…

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…

Given a compact interval $I \subseteq \mathbb{R}$, and a function $f$ that is a product of a nonzero polynomial with a Gaussian, it will be shown that the translates $\{ f(\cdot - \lambda) : \lambda \in \Lambda \}$ are complete in $C(I)$ if…

We consider self-adjoint semigroups $T_t = \exp(-tA)$ acting on $L^2(\Omega)$ and satisfying (generalised) Gaussian estimates, where $\Omega$ is a metric measure space of homogeneous type of dimension $d$. The aim of the article is to show…

Let $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{n})\in \mathbb{S}^{n-1}$ and $d\sigma$ denote the normalised Lebesgue measure on $\mathbb{S}^{n-1},~n\geq 2$. For functions $f_1, f_2,\dots,f_n$ defined on $\R$ consider the multilinear…

Consider the Dirichlet-Laplacian in $\Omega:= (0,L)\times (0,H)$ and choose another open set $\omega\subset \Omega$. The estimate $0<C_{\omega}\leq R_{\omega}(u):=\frac{\Vert u\Vert^{2}_{L^{2}(\omega)}}{\Vert u\Vert^{2}_{L^{2}(\Omega)}}\leq…

We study a family of double confluent Heun equations of the form $\mathcal E=0$, where $\mathcal L=\mathcal L_{\lambda,\mu,n}$ is a family of differential operators of order two. They depend on complex parameters $\lambda$, $\mu$, $n$. Its…

In this paper, we investigate the convergence of products of conditional expectation operators. We show that if $(\Omega,\cal{F},P)$ is a probability space that is not purely atomic, then divergent sequences of products of conditional…

Let $A$ be a complex square matrix, and write its polar decomposition as $A=U|A|$. For $0<\lambda<1$, the $\lambda$-Aluthge transform of $A$ is defined by $$ \Delta_\lambda(A)=|A|^\lambda U|A|^{1-\lambda}. $$ In 2007, Huang and Tam…

In previous work by Beer and Levi [8, 9], the authors studied the oscillation $\Omega (f,A)$ of a function $f$ between metric spaces $\langle X,d \rangle$ and $\langle Y,\rho \rangle$ at a nonempty subset $A$ of $X$, defined so that when $A…

Let $f\colon\mathbb{C}\to\mathbb{C}$ be a transcendental entire function. In 1989, Eremenko asked the following question concerning the set $I(f)$ of points that tend to infinity under iteration: can every point of $I(f)$ be joined to…

We obtain a complete description of closed ideals in weighted Lipschitz algebras $\Lambda_\omega$ of analytic functions on the unit disk satisfying the following condition $$\frac{|f(z)-f(w)|}{\omega(|z-w|)}=o(1)\qquad(as |z-w|…

It is shown that a function $f$ is a generalized Stieltjes function of order $\lambda>0$ if and only if $x^{1-\lambda}(x^{\lambda-1+k}f(x))^{(k)}$ is completely monotonic for all $k\geq 0$, thereby complementing a result due to Sokal.…