arXiv subjects

Paweł Pietrzycki

Publications and source records attributed to Paweł Pietrzycki.

13 recordsLinked to original sources

arXiv2026

Reduced Embry characterizations of quasinormal operators

We prove that, for any two integers $2\le m<n$, a bounded operator $A$ on a complex Hilbert space satisfying \[ A^{*k}A^k=(A^*A)^k, \qquad k\in\{m,n\}, \] is quasinormal. More generally, we show that the inequalities \[ A^{*m}A^m\ge(A^*A)^m \quad \text{and} \quad A^{*n}A^n\le(A^*A)^n \] already imply quasinormality. Together with the known single-exponent counterexamples, this completely resolves the reduced Embry problem for bounded operators: the identities $A^{*k}A^k=(A^*A)^k$, $k\in S\subseteq\{1,2,\ldots\}$, characterize quasinormality if and only if $S$ contains at least two distinct integers greater than one. The proof combines Hansen's inequality and the transformer inequality for weighted operator geometric means with an extremal argument for optimal power bounds.

math.FA↗

arXiv2026

Hyperrigidity III

In this paper, we study hyperrigidity for $C^*$-algebras. We will show that hyperrigidity can be expressed solely in terms of representations, without the need to involve general unital completely positive maps.

math.OA↗

arXiv2026

CPD $n$th roots of subnormal operators are subnormal

We investigate the $n$th root problem for bounded operators on a Hilbert space within the class of conditionally positive definite (CPD) operators determined by the Lévy--Khintchine formula. The class contains subnormal operators, complete hypercontractions of order $2$, and $3$-isometries. Our main result shows that if $T$ is a CPD operator such that $T^n$ is subnormal (resp., quasinormal, normal, or a $3$-isometry), then $T$ belongs to the corresponding class. This establishes the invariance of these classes under taking $n$th roots within the CPD class and extends several earlier results in operator theory. Furthermore, we provide characterizations of quasinormal and normal operators in terms of their CPD property and the structure of the representing triplet. Finally, we show that the classes of CPD and normaloid operators are distinct by means of both theoretical arguments and explicit examples.

math.FA↗

arXiv2026

Hyperrigidity I: singly generated commutative $C^*$-algebras

Although Arveson's hyperrigidity conjecture was recently resolved negatively by B. Bilich and A. Dor-On, the problem remains open for commutative $C^*$-algebras. Relatively few examples of hyperrigid sets are known in the commutative case. The main goal of this paper is to determine which sets of monomials in $t$ and $t^*$, where $t$ is a generator of a commutative unital $C^*$-algebra, are hyperrigid. We show that this class of hyperrigid sets has significant connections to other areas of functional analysis and mathematical physics. Moreover, we develop a topological approach based on weak and strong limits of normal (or subnormal) operators to characterize hyperrigidity tracing back to ideas of C. Kleski and L. G. Brown. Employing Choquet boundary techniques, we present examples that discuss the optimality of our results.

math.OA↗

arXiv2025

Hyperrigidity II: $R$-dilations, ideals and decompositions

We investigate the hyperrigidity of subsets of unital $C^*$-algebras annihilated by states (or, more generally, by completely positive maps). This is closely related to the concept of rigidity at $0$ introduced by G. Salomon, who studied hyperrigid subsets of Cuntz and Cuntz-Krieger algebras. The absence of the unit in a hyperrigid set allows for the existence of $R$-dilations with non-isometric $R$. The existence of such an $R$-dilation forces the state annihilating the hyperrigid set to be a character. Using a dilation-theoretic approach, we provide multiple equivalent criteria for hyperrigidity involving intertwining relations for representations, valid in both commutative and noncommutative settings. We develop structural models for such dilations via orthogonal decompositions into two or three components, determined by defect operators and generalized eigenspaces associated with underlying representations.

math.OA↗

arXiv2025

Cowen-Douglas operators and analytic continuation

In this paper, we study certain Banach spaces of analytic functions on which a left-invertible multiplication operator acts. It turns out that, under natural conditions, its left inverse is a Cowen-Douglas operator. We investigate how the analytic continuations of functions from an invariant subspace of this Cowen-Douglas operator relate to the spectrum of its restriction to that subspace.

math.FA↗

arXiv2022

On $n$th roots of bounded and unbounded quasinormal operators

In a recent paper [9], R. E. Curto, S. H. Lee and J. Yoon asked the following question: Let $T$ be a subnormal operator, and assume that $T^2$ is quasinormal. Does it follow that $T$ is quasinormal?. In [36] we answered this question in the affirmative. In the present paper, we will extend this result in two directions. Namely, we prove that both class A $n$th roots of bounded quasinormal operators and subnormal $n$th roots of unbounded quasinormal operators are quasinormal. We also show that a non-normal quasinormal operator having a quasinormal $n$th root has a non-quasinormal $n$th root.

math.FA↗

arXiv2021

On Cauchy dual operator and duality for Banach spaces of analytic functions

In this paper, two related types of dualities are investigated. The first is the duality between left-invertible operators and the second is the duality between Banach spaces of vector-valued analytic functions. We will examine a pair ($\mathcal{B},Ψ)$ consisting of a reflexive Banach spaces $\mathcal{B}$ of vector-valued analytic functions on which a left-invertible multiplication operator acts and an operator-valued holomorphic function $Ψ$. We prove that there exist a dual pair ($\mathcal{B}^\prime,Ψ^\prime)$ such that the space $\mathcal{B}^\prime$ is unitarily equivalent to the space $\mathcal{B}^*$ and the following intertwining relations hold \begin{equation*} \mathscr{L} \mathcal{U} = \mathcal{U}\mathscr{M}_z^* \quad\text{and}\quad \mathscr{M}_z\mathcal{U} = \mathcal{U} \mathscr{L}^*, \end{equation*} where $\mathcal{U}$ is the unitary operator between $\mathcal{B}^\prime$ and $\mathcal{B}^*$. In addition we show that $Ψ$ and $Ψ^\prime$ are connected through the relation\begin{equation*} \langle(Ψ^\prime( \bar{z}) e_1) (λ),e_2\rangle= \langle e_1,(Ψ( \bar{ λ}) e_2)(z)\rangle \end{equation*} for every $e_1,e_2\in E$, $z\in \varOmega$, $λ\in \varOmega^\prime$. If a left-invertible operator $T$ satisfies certain conditions, then both $T$ and the Cauchy dual operator $T^\prime$ can be modelled as a multiplication operator on reproducing kernel Hilbert spaces of vector-valued analytic functions $\mathscr{H}$ and $\mathscr{H}^\prime$, respectively. We prove that Hilbert space of the dual pair of $(\mathscr{H},Ψ)$ coincide with $\mathscr{H}^\prime$, where $Ψ$ is a certain operator-valued holomorphic function. Moreover, we characterize when the duality between spaces $\mathscr{H}$ and $\mathscr{H}^\prime$ obtained by identifying them with $\mathcal{H}$ is the same as the duality obtained from the Cauchy pairing.

math.FA↗

arXiv2021

Two-moment characterization of spectral measures on the real line

Kiukas, Lahti and Ylinen asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in a recent paper of the present authors. Let $T$ be a selfadjoint operator and $F$ be a Borel semispectral measure on the real line with compact support. For which positive integers $p< q$ do the equalities $T^k =\int_{\mathbb{R}} x^k F(dx)$, $k=p, q$, imply that $F$ is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if $p$ is odd and $q$ is even, and negative otherwise. The case $(p,q)=(1,2)$ closely related to intrinsic noise operator was solved by several authors including Kruszyński and de Muynck as well as Kiukas, Lahti and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on $C^*$-algebras is also solved.

math.FA↗

arXiv2020

Subnormal $n$th roots of quasinormal operators are quasinormal

In the recent paper, R. E. Curto, S. H. Lee, J. Yoon, asked the following question: Let $A$ be a subnormal operator, and assume that $A^2$ is quasinormal. Does it follow that $A$ is quasinormal? In this paper, we give an affirmative answer to this question. In fact, we prove more general result that subnormal $n$th roots of quasinormal operators are quasinormal.

math.FA↗

arXiv2020

Between the von Neumann inequality and the Crouzeix conjecture

A new concept of a deformed numerical range $W^ρ(T)$ is introduced. Here $T$ is a bounded linear operator or a matrix and $ ρ\in[1,+\infty)$ is a parameter. Each $W^ρ(T)$ is a closed convex set that contains the spectrum of $T$. Furthermore, $W^ρ(T)$ is decreasing with respect to $ ρ$ and $W^2(T)$ coincides with the numerical range. It is also shown that $W^ρ(T)$ is contained in the closed unit disc if and only if $T$ has a $ρ$ unitary dilation in the sense of Nágy-Foia\c s. The spectral constants of $W^ρ(T)$ are investigated, it is shown that it is monotone and continuous with respect to the parameter $ ρ$.

math.FA↗

arXiv2018

Reduced commutativity of moduli of operators

In this paper, we investigate the question of when the equations $A^{*s}A^{s}=(A^{*}A)^{s}$ for all $s \in S$, where $S$ is a finite set of positive integers, imply the quasinormality or normality of $A$. In particular, it is proved that if $S=\{p,m,m+p,n,n+p\}$, where $2\leq m < n$, then $A$ is quasinormal. Moreover, if $A$ is invertible and $S=\{m,n,n+m\}$, where $m \leq n$, then $A$ is normal. Furthermore, the case when $S=\{m,m+n\}$ and $A^{*n}A^n \leq (A^*A)^n$ is discussed.

math.FA↗

arXiv2015

The single equality $A^{n}A^n = (A^A)^n$ does not imply the quasinormality of weighted shifts on rootless directed trees

It is proved that each bounded injective bilateral weighted shift $W$ satisfying the equality $W^{*n}W^{n}=(W^{*}W)^{n}$ for some integer $n\geq 2$ is quasinormal. For any integer $n\geq 2$, an example of a bounded non-quasinormal weighted shift $A$ on a rootless directed tree with one branching vertex which satisfies the equality $A^{*n}A^{n}=(A^{*}A)^{n}$ is constructed. It is also shown that such an example can be constructed in the class of composition operators in $L^2$-spaces over $σ$-finite measure spaces.

math.FA↗