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Let T be a C_{\cdot 0}-contraction on a Hilbert space H and S be a non-trivial closed subspace of H. We prove that S is a T-invariant subspace of H if and only if there exists a Hilbert space D and a partially isometric operator \Pi :…

We develope a local theory for frames on finite dimensional Hilbert spaces. In particular, a bounded frame on a finite dimensional Hilbert space contains a subset which is a good Riesz basis for a percentage (arbitrarily close to one) of…

In this paper, we study regression problems over a separable Hilbert space with the square loss, covering non-parametric regression over a reproducing kernel Hilbert space. We investigate a class of spectral/regularized algorithms,…

We study the spectrum of operators $aT\in B(H)$ on a Hilbert space $H$ where $T$ is an isometry and $a$ belongs to a commutative $C^*$-subalgebra $C(X)\cong A\subseteq B(H)$ such that the formula $L(a)=T^*aT$ defines a faithful transfer…

Few years ago G\u{a}vru\c{t}a gave the notions of $K$-frame and atomic system for a linear bounded operator $K$ in a Hilbert space $\mathcal{H}$ in order to decompose $\mathcal{R}(K)$, the range of $K$, with a frame-like expansion. These…

Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of…

multiplication operator on a Hilbert space may be approximated with finite sections by choosing an orthonormal basis of the Hilbert space. Nonzero multiplication operators on $L^2$ spaces of functions are never compact and then such…

This work provides closed-form solutions and minimum achievable errors for a large class of low-rank approximation problems in Hilbert spaces. The proposed theorem generalizes to the case of bounded linear operators the previous results…

We give an elementary proof of a generalization of Bourgain and Tzafriri's Restricted Invertibility Theorem, which says roughly that any matrix with columns of unit length and bounded operator norm has a large coordinate subspace on which…

We discuss some basic properties of Lie group representations in rigged Hilbert spaces. In particular, we show that a differentiable representation in a rigged Hilbert space may be obtained as the projective limit of a family of continuous…

The Invariant Subset Problem on the Hilbert space is to know whether there exists a bounded linear operator $T$ on a separable infinite-dimensional Hilbert space $H$ such that the orbit $\{T^{n}x;\ n\ge 0\}$ of every non-zero vector $x\in…

In this note, we introduce a novel norm, termed the $t-$Berezin norm, on the algebra of all bounded linear operators defined on a reproducing kernel Hilbert space $\mathcal{H}$ as $$\|A\|_{t-ber} = \sup_{ \lambda, \mu \in \Omega} \left\{…

We characterize those bounded multilinear operators that factor through a Hilbert space in terms of its behavior in finite sequences. This extends a result, essentially due to S. Kwapie\'{n}, from the linear to the multilinear setting. We…

We study a specific family of symmetric norms on the algebra $\mathcal B(\mathcal H)$ of operators on a separable infinite-dimensional Hilbert space. With respect to each symmetric norm in this family the identity operator fails to attain…

The translation of an operator is defined by using conjugation with time-frequency shifts. Thus, one can define $\Lambda$-shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice $\Lambda$ in…

In this article a unified approach to iterative soft-thresholding algorithms for the solution of linear operator equations in infinite dimensional Hilbert spaces is presented. We formulate the algorithm in the framework of generalized…

For a given $\delta$, $0<\delta<1$, a Blaschke sequence $\sigma=\{\lambda_j\}$ is constructed such that every function $f$, $f\in H^\infty$, having $\delta<\delta_f=\inf_{\lambda\in\sigma}|f(\lambda)|\le\|f\|_\infty\le1$ is invertible in…

The idea that symmetries simplify or reduce the complexity of a system has been remarkably fruitful in physics, and especially in quantum mechanics. On a mathematical level, symmetry groups single out a certain structure in the Hilbert…

We observe that local embedding problems for certain Hardy and Bergman spaces of Dirichlet series are equivalent to boundedness of a class of composition operators. Following this, we perform a careful study of such composition operators…

We study expansions of Hilbert spaces with a bounded normal operator $T$. We axiomatize this theory in a natural language and identify all of its completions. We prove the definability of the adjoint $T^*$ and prove quantifier elimination…