arXiv subjects

Catalin Badea

Publications and source records attributed to Catalin Badea.

At least 19 recordsLinked to original sources

arXiv2026

On the abstract approach to spectral constants: a proof of the Clou\^atre--Ostermann--Ransford conjecture

Clou\^atre, Ostermann, and Ransford formulated an abstract version of Crouzeix's conjecture involving a bounded unital homomorphism from a uniform algebra into matrices and a unital antilinear map. They conjectured that contractivity of the associated symmetrised map forces the homomorphism to have norm at most two. We prove this conjecture, in fact without requiring the antilinear map to be contractive and for homomorphisms into the bounded operators on a Hilbert space. The proof combines positivity of real parts, an operator-valued Herglotz theorem, and the perturbation lemma of Lorist and Schwenninger used in the recent proof of Crouzeix's conjecture.

math.FA↗

arXiv2024

Schwarz-Pick type inequalities from an operator theoretical point of view

We use (versions of) the von Neumann inequality for Hilbert space contractions to prove several Schwarz-Pick inequalities. Specifically, we derive an alternate proof for a multi-point Schwarz-Pick inequality by Beardon and Minda, along with a generalized version for operators. Connections with model spaces and Peschl's invariant derivatives are established. Finally, Schwarz-Pick inequalities for analytic functions on polydisks and for higher order derivatives are discussed. An enhanced version of the Schwarz-Pick lemma, using the notion of distinguished variety, is obtained for the bidisk.

math.FA↗

arXiv2024

A note on the Huijsmans-de Pagter problem on finite dimensional ordered vector spaces

A classical problem posed in 1992 by Huijsmans and de Pagter asks whether, for every positive operator $T$ on a Banach lattice with spectrum $σ(T) = \{1\}$, the inequality $T \ge \operatorname{id}$ holds true. While the problem remains unsolved in its entirety, a positive solution is known in finite dimensions. In the broader context of ordered Banach spaces, Drnovšek provided an infinite-dimensional counterexample. In this note, we demonstrate the existence of finite-dimensional counterexamples, specifically on the ice cream cone and on a polyhedral cone in $\mathbb{R}^3$. On the other hand, taking inspiration from the notion of $m$-isometries, we establish that each counterexample must contain a Jordan block of size at least $3$.

math.FA↗

arXiv2023

Around Furstenberg's times $p$, times $q$ conjecture: times $p$-invariant measures with some large Fourier coefficients

For each integer $n\ge 1$, denote by $T_{n}$ the map $x\mapsto nx\mod 1$ from the circle group $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ into itself. Let $p,q\ge 2$ be two multiplicatively independent integers. Using Baire Category arguments, we show that generically a $T_{p}$-invariant probability measure $\mu$ on $\mathbb{T}$ with no atom has some large Fourier coefficients along the sequence $(q^n)_{n\ge 0}$. In particular, $(T_{q^{n}}\mu )_{n\ge 0}$ does not converges weak-star to the normalised Lebesgue measure on $\mathbb{T}$. This disproves a conjecture of Furstenberg and complements previous results of Johnson and Rudolph. In the spirit of previous work by Meiri and Lindenstrauss-Meiri-Peres, we study generalisations of our main result to certain classes of sequences $(c_n)_{n\ge 0}$ other than the sequences $(q^{n})_{n\ge 0}$, and also investigate the multidimensional setting.

math.DS↗

arXiv2022

Rochberg's abstract coboundary theorem revisited

Rochberg's coboundary theorem provides conditions under which the equation $(I-T)y = x$ is solvable in $y$. Here $T$ is a unilateral shift on Hilbert space, $I$ is the identity operator and $x$ is a given vector. The conditions are expressed in terms of Wold-type decomposition determined by $T$ and growth of iterates of $T$ at $x$. We revisit Rochberg's theorem and prove the following result. Let $T$ be an isometry acting on a Hilbert space $\mathcal{H}$ and let $x \in \mathcal{H}$. Suppose that $\sum_{k=0}^\infty k \| T^{*k} x \| < \infty$. Then $x$ is in the range of $(I-T)$ if (and only if) $\|\sum_{k= 0}^n T^k x \| = o(\sqrt{n}).$ When $T$ is merely a contraction, $x$ is a coboundary under an additional assumption. Some applications to $L^2$-solutions of the functional equation $f(x)-f(2x) = F(x)$, considered by Fortet and Kac, are given.

math.FA↗

arXiv2021

Hilbert space operators with two-isometric dilations

A bounded linear Hilbert space operator $S$ is said to be a $2$-isometry if the operator $S$ and its adjoint $S^*$ satisfy the relation $S^{*2}S^{2} - 2 S^{*}S + I = 0$. In this paper, we study Hilbert space operators having liftings or dilations to $2$-isometries. The adjoint of an operator which admits such liftings is characterized as the restriction of a backward shift on a Hilbert space of vector-valued analytic functions. These results are applied to concave operators (i.e., operators $S$ such that $S^{*2}S^{2} - 2 S^{*}S + I \le 0$) and to operators similar to contractions or isometries. Two types of liftings to $2$-isometries, as well as the extensions induced by them, are constructed and isomorphic minimal liftings are discussed.

math.FA↗

arXiv2020

High order isometric liftings and dilations

We show that a Hilbert space bounded linear operator has an $m$-isometric lifting for some integer $m\ge 1$ if and only if the norms of its powers grow polynomially. In analogy with unitary dilations of contractions, we prove that such operators also have an invertible $m$-isometric dilation. We also study $2$-isometric liftings of convex operators and $3$-isometric liftings of Foguel-Hankel operators.

math.FA↗

arXiv2019

Escaping a neighborhood along a prescribed sequence in Lie groups and Banach algebras

It is shown that Jamison sequences, introduced in 2007 by Badea and Grivaux ([C. Badea and S. Grivaux, Unimodular eigenvalues, uniformly distributed sequences and linear dynamics, Adv. Math. 211 (2007), no. 2, 766--793]), arise naturally in the study of topological groups with no small subgroups, of Banach or normed algebra elements whose powers are close to identity along subsequences, and in characterizations of (self-adjoint) positive operators by the accretiveness of some of their powers. The common core of these results is a description of those sequences for which non-identity elements in Lie groups or normed algebras escape an arbitrary small neighborhood of the identity in a number of steps belonging to the given sequence. Several spectral characterizations of Jamison sequences are given and other related results are proved.

math.FA↗

arXiv2019

Rigidity sequences, Kazhdan sets and group topologies on the integers

We study the relationships between three different classes of sequences (or sets) of integers, namely rigidity sequences, Kazhdan sequences (or sets) and nullpotent sequences. We prove that rigidity sequences are non-Kazhdan and nullpotent, and that all other implications are false. In particular, we show by probabilistic means that there exist sequences of integers which are both nullpotent and Kazhdan. Moreover, using Baire category methods, we provide general criteria for a sequence of integers to be a rigidity sequence. Finally, we give a new proof of the existence of rigidity sequences which are dense in $\mathbb{Z}$ for the Bohr topology, a result originally due to Griesmer.

math.DS↗

arXiv2018

The Cauchy dual and 2-isometric liftings of concave operators

We present some 2-isometric lifting and extension results for Hilbert space concave operators. For a special class of concave operators we study their Cauchy dual operators and discuss conditions under which these operators are subnormal. In particular, the quasinormality of compressions of such operators is studied.

math.FA↗

arXiv2018

Similarity problems, Folner sets and isometric representations of amenable semigroups

We revisit Sz.-Nagy's criteria for similarity of Hilbert space bounded linear operators to isometries or unitaries and present new ones. We also discuss counterparts of the Dixmier-Day theorem concerning bounded representations of amenable groups and semigroups. We highlight the role of Folner sets in similarity problems in both settings of unimodular, $σ$-compact, amenable groups and in discrete semigroups possessing the Strong Folner condition (SFC).

math.FA↗

arXiv2018

Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences

Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers $p_{1},\dots,p_{r}$ there exists a continuous probability measure $μ$ on the unit circle $\mathbb{T}$ such that \[ \inf_{k_{1}\ge 0,\dots,k_{r}\ge 0}|\widehat{μ}(p_{1}^{k_{1}}\dots p_{r}^{k_{r}})|>0. \] This results applies in particular to the Furstenberg set $F=\{2^{k}3^{k'}\,;\,k\ge 0,\ k'\ge 0\}$, and disproves a 1988 conjecture of Lyons inspired by Furstenberg's famous $\times 2$-$\times 3$ conjecture. We also estimate the modified Kazhdan constant of $F$ and obtain general results on rigidity sequences which allow us to retrieve essentially all known examples of such sequences.

math.DS↗

arXiv2018

Spectral sets and operator radii

We study different operator radii of homomorphisms from an operator algebra into $B(H)$ and show that these can be computed explicitly in terms of the usual norm. As an application, we show that if $Ω$ is a $K$-spectral set for a Hilbert space operator, then it is a $M$-numerical radius set, where $M=\frac{1}{2}(K+K^{-1})$. This is a counterpart of a recent result of Davidson, Paulsen and Woerdeman. More general results for operator radii associated with the class of operators having $ρ$-dilations in the sense of Sz.-Nagy and Foias are given. A version of a result of Drury concerning the joint numerical radius of non-commuting $n$-tuples of operators is also obtained.

math.FA↗

arXiv2018

Sets of integers determined by operator-theoretical properties: Jamison and Kazhdan sets in the group $\mathbb{Z}$

The aim of this partly expository paper is to present and discuss two classes of sets of integers (Jamison and Kazhdan sets) whose definition and/or properties are determined or inspired by operator-theoretical properties. Jamison sets first appeared in the study of the relationship between the growth of the sequence of norms of iterates of a bounded linear operator on a separable Banach space and the size of its unimodular point spectrum. Kazhdan subsets of $\mathbb{Z}$ are particular cases of Kazhdan sets in general topological groups, which are especially important as they appear in the definition of Property (T). This paper is also intended as a companion to the authors' paper [C.Badea, S.Grivaux, Kazhdan sets in groups and equidistribution properties, \emph{J. Funct. Anal.} \textbf{273} (2017), p. 1931 -- 1969], which undertakes a study of Kazhdan subsets of some classical groups without Property (T). We present here in detail the case of the group $\mathbb{Z}$, which is one of the most natural examples of groups without Property (T), and which may be useful to build an intuition of some of the main results of [C.Badea, S.Grivaux, Kazhdan sets in groups and equidistribution properties, op. cit.]. Also, the proofs in the case of the group $\mathbb{Z}$ rely solely on tools from basic operator theory and harmonic analysis. Some crucial links between Jamison and Kazhdan sets in $\mathbb{Z}$ are exhibited, and many examples are given.

math.FA↗

arXiv2017

Harnack and Shmul'yan pre-order relations for Hilbert space contractions

We study the behavior of some classes of Hilbert space contractions with respect to Harnack and Shmul'yan pre-orders and the corresponding equivalence relations. We give some conditions under which the Harnack equivalence of two given contractions is equivalent to their Shmul'yan equivalence and to the existence of an arc joining the two contractions in the class of operator-valued contractive analytic functions on the unit disc. We apply some of these results to quasi-isometries and quasi-normal contractions, as well as to partial isometries for which we show that their Harnack and Shmul'yan parts coincide. We also discuss an extension, recently considered by S.~ter~Horst [\emph{J. Operator Th. 72(2014), 487--520}], of the Shmul'yan pre-order from contractions to the operator-valued Schur class of functions. In particular, the Shmul'yan-ter Horst part of a given partial isometry, viewed as a constant Schur class function, is explicitly determined.

math.FA↗

arXiv2017

Kazhdan sets in groups and equidistribution properties

Using functional and harmonic analysis methods, we study Kazhdan sets in topological groups which do not necessarily have Property (T). We provide a new criterion for a generating subset $Q$ of a group $G$ to be a Kazhdan set; it relies on the existence of a positive number $\varepsilon$ such that every unitary representation of $G$ with a $(Q,\varepsilon )$-invariant vector has a finite dimensional subrepresentation. Using this result, we give an equidistribution criterion for a generating subset of $G$ to be a Kazhdan set. In the case where $G=\mathbb{Z}$, this shows that if $(n_{k})_{k\ge 1}$ is a sequence of integers such that $(e^{2iπθn_{k}})_{k\ge 1}$ is uniformly distributed in the unit circle for all real numbers $θ$ except at most countably many, then $\{n_{k}\,;\,k\ge 1\}$ is a Kazhdan set in $\mathbb{Z}$ as soon as it generates $\mathbb{Z}$. This answers a question of Y. Shalom from [B.~Bekka, P.~de la~Harpe, A.~Valette, Kazhdan's property (T), Cambridge Univ. Press, 2008]. We also obtain characterizations of Kazhdan sets in second countable locally compact abelian groups, in the Heisenberg groups and in the group $\textrm{Aff}_{+}(\mathbb{R})$. This answers in particular a question from [B.~Bekka, P.~de la~Harpe, A.~Valette, Kazhdan's property (T), op. cit.].

math.GR↗

arXiv2017

Quantified asymptotic behaviour of Banach space operators and applications to iterative projection methods

We present an extension of our earlier work [Ritt operators and convergence in the method of alternating projections, J. Approx. Theory, 205:133-148, 2016] by proving a general asymptotic result for orbits of an operator acting on a reflexive Banach space. This result is obtained under a condition involving the growth of the resolvent, and we also discuss conditions involving the location and the geometry of the numerical range of the operator. We then apply the general results to some classes of iterative projection methods in approximation theory, such as the Douglas-Rachford splitting method and, under suitable geometric conditions either on the ambient Banach space or on the projection operators, the method of alternating projections.

math.FA↗

arXiv2016

Classes of contractions and Harnack domination

Several properties of the Harnack domination of linear operators acting on Hilbert space with norm less or equal than one are studied. Thus, the maximal elements for this relation are identified as precisely the singular unitary operators, while the minimal elements are shown to be the isometries and the adjoints of isometries. We also show how a large range of properties (e.g. convergence of iterates, peripheral spectrum, ergodic properties) are transfered from a contraction to one that Harnack dominates it.

math.FA↗