Related papers: Negative powers of Hilbert-space contractions

Let $(E,\mathcal E,\mu)$ be a measure space and $G\colon E\times E\to [0,\infty]$ be measurable. Moreover, let $\mathcal F\!_{ui}$ denote the set of all $q\in\mathcal E^+$ (measurable numerical functions $q\ge 0$ on $E$) such that…

Let $r_A(T)$ denote the $A$-spectral radius of an operator $T$ which is bounded with respect to the seminorm induced by a positive operator $A$ on a complex Hilbert space $\mathcal{H}$. In this paper, we aim to establish some $A$-spectral…

We discuss the two closely related, but different concepts of weak and almost weak stability for the powers of a contraction on a separable Hilbert space. Extending Halmos' and Rohlin's theorems in ergodic theory as a model, we show that…

This is a continuation of our previous paper. We consider a certain order-like relation for positive operators on a Hilbert space. This relation is defined by using the Jensen inequality with respect to the square-root function. We show…

We prove sufficient conditions for Hausdorff convergence of the spectra of sequences of closed operators defined on varying Hilbert spaces and converging in norm-resolvent sense, i.e. $\|J_\varepsilon(1+A_\varepsilon)^{-1} -…

Let $T$ be a bounded linear operator on a Banach space $X$ satisfying $\|T^n\|/n \to 0$. We prove that $T$ is uniformly ergodic if and only if the one-sided ergodic Hilbert transform $H_Tx:= \lim_{n\to\infty} \sum_{k=1}^n k^{-1}T^k x$…

A truncated Toeplitz operator is the compression $A_{\phi}:\K_{\Theta} \to \K_{\Theta}$ of a Toeplitz operator $T_{\phi}:H^2\to H^2$ to a model space $\K_{\Theta} := H^2 \ominus \Theta H^2$. For $\Theta$ inner, let $\T_{\Theta}$ denote the…

In this paper we study traces of an integral operator on two orthogonal subspaces of a $L^2$ space. One of the two traces is shown to be zero. Also, we prove that the trace of the operator on the second subspace is nonnegative. Hence, the…

We characterize the spectrum of positive linear operators $T:X \to Y$, where $X$ and $Y$ are complex Banach function spaces with unit $1$, having finite rank and a partition of unity property. Then all the points in the spectrum are…

For a Gaussian measure on a separable Hilbert space with covariance operator $C$, we show that the family of conditional measures associated with conditioning on a closed subspace $S^{\perp}$ are Gaussian with covariance operator the short…

A commuting triple of Hilbert space operators $(A,B,P)$, for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set, is called a \textit{tetrablock-contraction} or simply an $\mathbb E$-\textit{contraction}, where \[ \mathbb…

Let $u_1,\ldots,u_n$ be unitary operators on a Hilbert space $H$. We study the norm $$\left\|\sum^{i=n}_{i=1} u_i \otimes \bar u_i\right\|\leqno (1)$$ of the operator $\sum u_i \otimes \bar u_i$ acting on the Hilbertian tensor product…

This is an introduction to the algebras $A\subset B(H)$ that the linear operators $T:H\to H$ can form, once a complex Hilbert space $H$ is given. Motivated by quantum mechanics, we are mainly interested in the von Neumann algebras, which…

A bounded linear operator $A$ on a Hilbert space $\mathcal{H}$ is posinormal if there exists a positive operator $P$ such that $AA^{*} = A^{*}PA$. We show that if $A$ is posinormal with closed range, then $A^n$ is posinormal and has closed…

We constructed a unitary semigroup $(e^{tA})_{t \geq 0}$ on a Hilbert space and an orthogonal projection $P$ such that the limit $\lim_{n \to \infty} [ e^{\frac{t}{n}A}P ]^n$ does not exist strongly. A similar example with a positive…

Let $\mathcal{H}$ be a linear space equipped with an indefinite inner product $[\cdot, \cdot]$. Denote by $\mathcal{F}_{++}=\{f\in\mathcal{H} \ : \ [f,f]>0\}$ the nonlinear set of positive vectors in $\mathcal{H}$. We demonstrate that the…

We prove for an arbitrary complex $^*$-algebra $A$ that every topologically irreducible $^*$-representation of $A$ on a Hilbert space is finite dimensional precisely when the Lebesgue decomposition of representable positive functionals over…

Let E_lambda be a Hilbert space, whose elements are functions spanned by the eigenfunctions of the Laplace-Beltrami operator associated with an eigenvalue lambda>0. The norm of elements in this space is given by the Petersson inner product.…

We study positive Toeplitz operators on the Bergman spaces having the fast decreasing weight functions in a certain class. We give the characterizations for the boundedness and compactness of Toeplitz operators in terms of their Berezin…

The Riesz-Markov theorem identifies any positive, finite, and regular Borel measure on the complex unit circle with a positive linear functional on the continuous functions. By the Weierstrass approximation theorem, the continuous functions…