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In this paper we study subspaces which are invariant under squares and cubes (separately as well as jointly) of unicellular backward weighted shift operators on a separable Hilbert space. The finite-dimensional subspaces are characterized…

Representations by linear integral operators on $L_p$ spaces over measure spaces are investigated for the polynomial covariance type commutation relations and more general two-sided generalizations of covariance commutation relations…

The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…

If $T$ is a semibounded self-adjoint operator in a Hilbert space $(H, \, (\cdot , \cdot))$ then the closure of the sesquilinear form $(T \cdot , \cdot)$ is a unique Hilbert space completion. In the non-semibounded case a closure is a…

Let T be a bounded operator on a Hilbert space H, and F = {f_j: j in J} an at most countable set of vectors in H. In this note, we characterize the pairs {T, F} such that {T^n f: f in F, n in I} form a frame of H, for the cases of I = N_0…

A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier-Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical…

Let $\{T(t)\}_{t\ge 0}$ be a $C_0$-semigroup on a separable Hilbert space $H$. We characterize that $T(t)$ is an $m$-isometry for every $t$ in terms that the mapping $t\in \Bbb R^+ \rightarrow \|T(t)x\|^2$ is a polynomial of degree less…

In the present paper we introduce the class of slice-polynomial functions: slice regular functions {defined over the quaternions, outside the real axis,} whose restriction to any complex half-plane is a polynomial. These functions naturally…

This paper proves two theorems. The first of these simplifies and lends clarity to the previous characterizations of the invariant subspaces of $S$, the operator of multiplication by the coordinate function $z$, on…

Let K be the product O(n_1) x O(n_2) x ... x O(n_r) of orthogonal groups. Let V the r-fold tensor product of defining representations of each orthogonal factor. We compute a stable formula for the dimension of the K-invariant algebra of…

For a large class of unitarily invariant reproducing kernel functions $K$ on the unit ball $\mathbb B_d$ in $\mathbb C^d$, we characterize the $K$-inner functions on $\mathbb B_d$ as functions admitting a suitable transfer function…

Let $H$ be a separable Hilbert space with a fixed orthonormal basis. Let $\mathbb B^{(k)}(H)$ denote the set of operators, whose matrices have no more than $k$ non-zero entries in each line and in each column. The closure of the union (over…

We introduce \textit{singular value functions} for C\(^*\)-algebras, generalizing the singular values of compact operators on Hilbert spaces. We also establish several fundamental properties of these singular value functions and present…

Let ${\cal C}$ be an algebraic space curve defined parametrically by ${\cal P}(t)\in {\Bbb K}(t)^{n},\,n\geq 2$. In this paper, we introduce a polynomial, the T--function, $T(s)$, which is defined by means of a univariate resultant…

The aim of this note is to exploit a new relationship between additive combinatorics and the geometry of monomial projective curves. We associate to a finite set of non-negative integers $A=\{a_1,\cdots, a_n\}$ a monomial projective curve…

A commuting triple of Hilbert space operators $(A,S,P)$ is said to be a \textit{$\mathbb{P}$-contraction} if the closed pentablock $\overline{\mathbb P}$ is a spectral set for $(A,S,P)$, where \[ \mathbb{P}:=\left\{(a_{21}, \mbox{tr}(A_0),…

In this paper we initiate the study of a fundamental yet untapped random model of non-selfadjoint, bounded linear operators acting on a separable complex Hilbert space. We replace the weights $w_n=1$ in the classical unilateral shift $T$,…

Motivated by Stanley's conjecture on the multiplication of Jack symmetric functions, we prove a couple of identities showing that skew Jack symmetric functions are semi-invariant up to translation and rotation of a $\pi$ angle of the skew…

We consider a Hilbert space that is a product of a finite number of Hilbert spaces and operators that are represented by "componental operators" acting on the Hilbert spaces that form the product space. We attribute operatorial properties…

An operatorial polynomial polyhedron is a set of the form $$B_{\delta}(\mathcal{B}(\mathcal{H}))=\{X\in \mathcal{B}(\mathcal{H})^d : \Vert\delta(X)\Vert<1\}$$ where $\mathcal{B}(\mathcal{H})$ denotes the space of bounded operators on a…