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Marek Ptak

Publications and source records attributed to Marek Ptak.

At least 19 recordsLinked to original sources
arXiv2026

Unitarily invariant norms

This survey paper provides a comprehensive study of unitarily invariant norms on the algebra of $n \times n$ matrices. This investigation leads naturally to the theory of symmetric gauge functions, a class of norms on $\mathbb{R}^n$ characterized by invariance and monotonicity properties. We develop the necessary framework by examining absolute and monotone norms and establishing their equivalence, thereby offering additional insight into the classical Hardy-Littlewood-P\'{o}lya theorem on majorization. The theory of majorization is further explored through its connections with doubly stochastic matrices, convexity, and fundamental results such as the Birkhoff and Rad\'{o} theorems, as well as K\"{o}nig's theorem on term rank and line rank. We also study weak majorization and derive a characterization that plays a crucial role in proving the monotonicity of symmetric gauge functions. On the spectral side, we review key results in matrix analysis, including the Courant-Fischer min-max theorem, the Cauchy interlacing theorem, and Ky Fan's majorization theorem, along with a weak subadditivity result for singular values of arbitrary matrices. These ingredients culminate in a detailed proof of von Neumann's characterization of unitarily invariant norms, which provides a complete and elegant description of this class of norms. Some illustrative examples, as well as the Ky Fan domination principle as an application, are also presented.

math.FA↗
arXiv2026

The conjugate orbit of a unitary operator

This paper discusses various aspects of the collection of unitary operators $CUC$, where $U$ is a fixed unitary operator on a complex Hilbert space $\mathcal{H}$ and $C$ varies over the set of all conjugations on $\mathcal{H}$ (antilinear, isometric, involutions). We call this class of unitary operators, the {\em conjugate orbit }of $U$ and denote it by $\mathfrak{O}_c(U)$. We will see that $U^{*}$, the Hilbert space adjoint of $U$, always belongs to $\mathfrak{O}_c(U)$, while $U$ belongs to $\mathfrak{O}_c(U)$ only when $U$ is unitarily equivalent to $U^{*}$, making $U$ a member of $\mathfrak{O}_c(U)$ an uncommon event. We completely describe the conjugate orbit of the classical bilateral shift and discuss when a unitary multiplication operator on the classical Lebesgue space of the unit circle belongs to this conjugate orbit. We also broaden this discussion to include the bilateral shifts of higher multiplicity which, via unitary equivalence, makes connections to other interesting unitary operators such as the translation and dilation operators on the Lebesgue space of the real line. Finite unitary matrices provide us with a rich source of examples of conjugate unitary orbits to discuss. In particular, we determine which diagonal matrices, if any, belong to the conjugate orbit of a fixed unitary matrix. Closely related to the finite unitary matrices are the diagonalizable unitary operators with respect to some, possibly infinite, orthonormal basis. We give a large class of variations of these unitary operators that belong to the conjugate orbit and establish a connection to the classical Fourier--Plancherel and Hilbert transforms. Finally, we develop a model for a unitary operator using real Hilbert spaces and use it to describe the conjugate orbit as well as revisit some of our previous discussions in another light.

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arXiv2024

Conjugations of Unitary operators, I

If $U$ is a unitary operator on a separable complex Hilbert space $\mathcal{H}$, an application of the spectral theorem says there is a conjugation $C$ on $\mathcal{H}$ (an antilinear, involutive, isometry on $\mathcal{H}$) for which $ C U C = U^{*}.$ In this paper, we fix a unitary operator $U$ and describe all of the conjugations $C$ which satisfy this property. As a consequence of our results, we show that a subspace is hyperinvariant for $U$ if and only if it is invariant for any conjugation $C$ for which $CUC = U^{*}$.

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arXiv2024

Conjugations of Unitary Operators, II

For a given unitary operator $U$ on a separable complex Hilbert space $\h$, we describe the set $\mathscr{C}_{c}(U)$ of all conjugations $C$ (antilinear, isometric, and involutive maps) on $\h$ for which $C U C = U$. As this set might be empty, we also show that $\mathscr{C}_{c}(U) \not = \varnothing$ if and only if $U$ is unitarily equivalent to $U^{*}$.

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arXiv2022

Shift invariance and reflexivity of compressions of multiplication operators

The property of being shift invariant and being reflexive or transitive in the case of the space of (asymmetric) truncated Toeplitz operators, and the space of (asymmetric) dual truncated operators is investigated. Most of the results obtained are new even for the symmetric case. A characterization of asymmetric dual truncated Toeplitz operators is also given.

math.FA↗
arXiv2021

Square roots of some classical operators

In this paper we give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical operators included in this discussion are the square of the unilateral shift, the Volterra operator, certain compressed shifts, the unilateral shift plus its adjoint, the Hilbert matrix, and the Cesàro operator.

math.FA↗
arXiv2020

Intertwining property for compressions of multiplication operators

Following Beurling's theorem the natural compressions of the multiplication operator in the classical $L^2$ space are compressions to model spaces and to their orthogonal complements. Two possibly different model spaces are considered hence asymmetric truncated Toeplitz and asymmetric dual truncated Toeplitz operators are investigated. The main purpose of the paper is to characterize operators which intertwine compressions of the unilateral shift.

math.FA↗
arXiv2020

Interpolating with outer functions

The classical theorems of Mittag-Leffler and Weierstrass show that when $\{λ_n\}$ is a sequence of distinct points in the open unit disk $\D$, with no accumulation points in $\D$, and $\{w_n\}$ is any sequence of complex numbers, there is an analytic function $ϕ$ on $\D$ for which $ϕ(λ_n) = w_n$. A celebrated theorem of Carleson \cite{MR117349} characterizes when, for a bounded sequence $\{w_n\}$, this interpolating problem can be solved with a bounded analytic function. A theorem of Earl \cite{MR284588} goes further and shows that when Carleson's condition is satisfied, the interpolating function $ϕ$ can be a constant multiple of a Blaschke product. In this paper, we explore when the interpolating $ϕ$ can be an outer function. We then use our results to refine a result of McCarthy \cite{MR1065054} and explore the common range of the co-analytic Toeplitz operators on a model space.

math.CV↗
arXiv2020

Asymmetric truncated Toeplitz operators and conjugations

Truncated Toeplitz operators in a model space are C--symmetric with respect to a natural conjugation in that space. We show that this and another conjugation associated to an orthogonal decomposition possess unique properties and we study their relations with asymmetric truncated Toeplitz operators in terms of C--symmetry. New connections with Hankel operators are established through this approach.

math.FA↗
arXiv2019

Conjugations in $L^2$ and their invariants

Conjugations in space $L^2$ of the unit circle commuting with multiplication by $z$ or intertwining multiplications by $z$ and $\bar z$ are characterized. We also study their behaviour with respect to the Hardy space, subspaces invariant for the unilateral shift and model spaces.

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arXiv2019

Invertibility, Fredholmness and kernels of dual truncated Toeplitz operators

Asymmetric dual truncated Toeplitz operators acting between the orthogonal complements of two (eventually different) model spaces are introduced and studied. They are shown to be equivalent after extension to paired operators on $L^2(\mathbb T) \oplus L^2(\mathbb T)$ and, if their symbols are invertible in $L^\infty(\mathbb T)$, to asymmetric truncated Toeplitz operators with the inverse symbol. Relations with Carleson's corona theorem are also established. These results are used to study the Fredholmness, the invertibility and the spectra of various classes of dual truncated Toeplitz operators.

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arXiv2019

Conjugations in $L^2(\mathcal{H})$

Conjugations commuting with $\mathbf{M}_z$ and intertwining $\mathbf{M}_z$ and $\mathbf{M}_{\bar z}$ in $L^2(\mathcal{H})$, where $\mathcal{H}$ is a Hilbert space, are characterized. We also investigate which of them leave invariant the whole Hardy space $H^2(\mathcal{H})$ or a model space $K_Θ=H^2(\mathcal{H})\ominusΘH^2(\mathcal{H})$, where $Θ$ is a pure operator valued inner function.

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arXiv2019

$C$--normal operators

A new class of operators, larger than $C$-symmetric operators and different than normal one, named $C$--normal operators is introduced. Basic properties are given. Characterizations of this operators in finite dimensional spaces using a relation with conjugate normal matrices are presented.

math.FA↗
arXiv2017

Generalized multipliers for left-invertible analytic operators and its applications to commutant and reflexivity

We introduce generalized multipliers for left-invertible analytic operators. We show that they form a Banach algebra and characterize the commutant of such operators in its terms. In the special case, we describe the commutant of balanced weighted shift only in terms of its weights. In addition, we prove two independent criteria for reflexivity of weighted shifts on directed trees.

math.FA↗
arXiv2017

Weighted shifts on directed trees. Their multiplier algebras, reflexivity and decompositions

We study bounded weighted shifts on directed trees. We show that the set of multiplication operators associated with an injective weighted shift on a rooted directed tree coincides with the WOT/SOT closure of the set of polynomials of the weighted shift. From this fact we deduce reflexivity of those weighted shifts on rooted directed trees whose all path-induced spectral-like radii are positive. We show that weighted shifts with positive weights on rooted directed trees admit a Wold-type decomposition. We prove that the pairwise orthogonality of the factors in the decomposition is equivalent to the weighted shift being balanced.

math.FA↗
arXiv2016

Characterization of truncated Toeplitz operators by conjugations

Truncated Toeplitz operators are C--symmetric with respect to the canonical conjugation given on an appropriate model space. However, by considering only one conjugation one cannot characterize truncated Toeplitz operators. It will be proved, for some classes of inner functions and the model spaces connected with them, that if an operator on a model space is C--symmetric for a certain family of conjugations in the model space, then is has to be truncated Toeplitz. A characterization of classical Toeplitz operators is also presented in terms of conjugations.

math.FA↗
arXiv2016

Characterizations of asymmetric truncated Toeplitz operators

The aim of this paper is to investigate asymmetric truncated Toeplitz operators with $L^2$ symbols between two different model spaces given by inner functions such that one divides the other. Characterizations of these operators are given in terms of rank two operators. A description of the class of symbols for which the corresponding asymmetric truncated Toeplitz operator is equal to the zero operator is also given.

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