This article, prepared for the Problem List of the Bench Math Session 2025, discusses the (anti)holomorphic rigidity phenomenon of Kobayashi and Carathéodory isometries. In particular, we provide an overview of this phenomenon, summarize several known results, and pose some open questions.
We shall discuss the characterization of cyclicity of polynomials in Dirichlet-type spaces in the Euclidean unit ball of Cn ${\mathbb{C}}^{n}$ which remains an open problem.
This note presents a problem submitted for the Bench Math Session 2025. The problem concerns the understanding of weighted Szeg & odblac; kernels in relation to auxiliary functions considered by Nehari. In particular, we pose a question about the zeros of the weighted Szeg & odblac; kernel, which lies within the scope of the classical Nehari problem studied in the literature.
We introduce the reader to partial isometries and their properties. In particular, we consider the Gau-Wang-Wu conjecture from 2016: When is the numerical range of a partial isometry circular? In order to describe a simple proof of a special case of the conjecture, we discuss basic properties of Blaschke products, compressions of the shift operator and some classical geometric methods.
In this paper, we are interested in studying surjective isometries of C*-algebras with an emphasis on their complex spectrum. We do not require the isometries to be linear nor the C*-algebras to be unital. We first characterize such isometries in terms of a Jordan & lowast;-isomorphism, a central projection, and a unitary element of the multiplier algebra, following a long line of work that began with the Banach-Stone theorem. In our main result, we then establish, for a wide class of surjective isometries, a precise connection between the complex spectrum of the isometry and the classical spectrum of the associated Jordan & lowast;-isomorphism. Finally, we turn our attention to periodic surjective isometries and provide several examples that illustrate the range of possibilities that can occur for the complex spectrum of the isometry and classical spectrum of the Jordan & lowast;-isomorphism.
We provide a brief introduction to the spectral theory of Toeplitz operators on Hardy spaces built upon L p-spaces and their abstract generalisations.
This note presents a question regarding the relationship between approximation by strongly incomplete polynomials on compact subsets K of C $\mathbb{C}$ and the logarithmic capacity of K.
In this short paper we will discuss recent advances on the problem of characterizing the boundedness of the composition operator acting on the Bergman spaces A(beta)(2)(D-2) whenever the self map phi of the bidisc is induced by Rational Inner Functions. The problem stated here is submitted as part of the Problem List of the Young Researchers Workshop in Complex Analysis and Operator Theory "The Bench Math Session 2025", organized in Jagiellonian University of Krakow at 10th-11th February 2025.
We present an old problem in Geometric Function Theory: characterizing the domains Ω⊂C ${\Omega}\subset \mathbb{C}$ for which every holomorphic map from the unit disk into Ω must belong to a specific function space X. We will provide some generalities about this problem, although the main aim is to delve into the geometric characterizations for several classical spaces, such as the Bloch space B $\mathcal{B}$ , the spaces of analytic functions of bounded mean oscillation BMOA, the Nevanlinna class N, the Smirnov class N +, the Hardy spaces H p, and the weighted Bergman spaces Aαp ${A}_{\alpha }^{p}$ . This work synthesizes a number of seminal results by several authors, aiming to provide a unified introduction to this classical but active topic.
In this work, density of the algebra A(D) $\mathcal{A}\left(D\right)$ in H(b) spaces of finitely connected planar domains and the boundedness of composition operators on these function spaces are studied. Density of the algebra is considered when the defining function b is a non-extreme point of the unit ball of H ∞(D). In the last part boundedness of composition operators on H(b) spaces is considered and as well as a generalization of the unit disk case is given, the boundedness of composition operators with generalized Blaschke symbols over finitely connected domains is characterized.
These notes are based on a mini-course given at the ACOTCA conference 2025. The goal is to present full proofs of the first two key results regarding hypercyclic Toeplitz operators, in a way that is accessible to beginners.
In this paper, we revisit the Riemann-Liouville analytic semigroup. In particular, we completely characterize the membership in the Schatten class S r ${\mathcal{S}}<^>{r}$ on L 2(0, 1), as well as the membership in the class of nuclear operators on L p (0, 1), p >= 1, and the membership in the ideal of absolutely r-summing operators for any r >= 1.
For 0 < p ⩽ ∞ 0\lt p\leqslant \infty and 0 < r ⩽ ∞ 0\lt r\leqslant \infty , the space M p , r {{\mathfrak{M}}}_{p,r} of (coefficient) multipliers from ℓ p {\ell }^{p} to ℓ r {\ell }^{r} is completely characterized. This is elementary in most instances. The interesting case 0 < r < p < ∞ 0\lt r\lt p\lt \infty requires more effort, and it is shown that a sequence of complex numbers belongs to M p , r {{\mathfrak{M}}}_{p,r} if and only if the sequence of their absolute values has a nonincreasing rearrangement ( h 0 , h 1 , h 2 , … ) \left({h}_{0},{h}_{1},{h}_{2},\ldots ) satisfying ∑ k = 0 ∞ ( k + 1 ) ( p − r ) ⁄ p ( h k r − h k + 1 r ) 1 ⁄ r < ∞ . {\left(\mathop{\sum }\limits_{k=0}^{\infty }{\left(k+1)}^{\left(p-r)/p}\left({h}_{k}^{r}-{h}_{k+1}^{r})\right)}^{1/r}\lt \infty . In that case, the expression on the left is the norm of the multiplier, and it is a compact operator. Further upper and lower bounds are given for the multiplier norm.
For 0 satisfying ((infinity)& sum;(k=0)(k+1)((p-r)& frasl;p)(h (R)(k)-h (R)(k+1))(1/r) < infinity In that case, the expression on the left is the norm of the multiplier, and it is a compact operator. Further upperand lower bounds are given for the multiplier norm
In this article, we initially study when an anti-linear Toeplitz operator is in the commutant of a composition operator. Primarily, we investigate weighted composition operators Wg,ψ{W}_{g,\psi } commuting with complex symmetric weighted composition operators Wf,φ{W}_{f,\varphi } on the Hardy space H2(D){H}^{2}\left({\mathbb{D}}). In particular, we give the descriptions of the symbols gg and ψ\psi such that the inducing weighted composition operator Wg,ψ{W}_{g,\psi } commutes with the complex symmetric weighted composition operator Wf,φ{W}_{f,\varphi } with the conjugation J{\mathcal{J}}. Furthermore, we subsequently demonstrate that these weighted composition operators are normal and complex symmetric in accordance with the properties of the fixed point of the associated symbol φ\varphi .
This article extends topics about linear algebra and operator theoretic linear transformations on complex vector spaces to those on bicomplex spaces. For example, Definition 3 for the first time defines algebraically idempotent vector spaces, which generalizes the standard definition of a vector space and which includes bicomplex vector spaces as a special case, along with its dimension and its basis in terms of a corresponding vectorial idempotent representation. The article also shows how an n×nn\times n bicomplex matrix’s idempotent representation leads to a bicomplex Jordan form and a description of its bicomplex invariant subspace lattice diagram. Similarly, in a new way, the article rigorously defines “bicomplex Banach and Hilbert” spaces, and then it expands, for the first time, the theory of compact operators on complex Banach spaces to those on bicomplex Banach spaces. In these ways, the article indicates that the idempotent representation extends complex linear algebra and operator theory in a surprisingly generalized and straightforward way to vector space results with bicomplex and multicomplex scalars.
We study the general properties, point spectrum and spectrum of a weighted composition operator Wm,φ{W}_{m,\varphi } with elliptic symbol φ\varphi on the unit ball BN{{\mathbb{B}}}_{N} of CN{{\mathbb{C}}}^{N}, and general weight m∈Hol(BN)m\in {\rm{Hol}}\left({{\mathbb{B}}}_{N}). We give a complete description of the spectra in the majority of cases, and we provide inclusions in general.
A major open question in the theory of Toeplitz operator on the Bergman space of the unit disk of the complex plane is the complete characterization of the set of all Toeplitz operators that commute with a given operator. Researchers showed that when a sum S = T e i m theta f + T e i l theta g S={T}_{{e}<^>{im\theta }f}+{T}_{{e}<^>{il\theta }g} , where f f and g g are radial functions, commutes with a sum T = T e i p theta r ( 2 M + 1 ) p + T e i s theta r ( 2 N + 1 ) s T={T}_{{e}<^>{ip\theta }{r}<^>{\left(2M+1)p}}+{T}_{{e}<^>{is\theta }{r}<^>{\left(2N+1)s}} , then S S must be of the form S = c T S=cT , where c c is a constant. In this article, we will replace r ( 2 M + 1 ) p {r}<^>{\left(2M+1)p} and r ( 2 N + 1 ) s {r}<^>{\left(2N+1)s} with r n {r}<^>{n} and r d {r}<^>{d} , where n n and d d are in N {\mathbb{N}} , and we will show that the same result holds.
This article studies the level set Crouzeix (LSC) conjecture, which is a weak version of Crouzeix's conjecture that applies to finite compressions of the shift. Among other results, this article establishes the LSC conjecture for several classes of 3 x 3 3\times 3 , 4 x 4 4\times 4 , and 5 x 5 5\times 5 matrices associated with compressions of the shift via a geometric analysis of their numerical ranges. This study also establishes Crouzeix's conjecture for several classes of nilpotent matrices whose studies are motivated by related compressions of shifts.
Abstract In this article, all sequences u {\boldsymbol{u}} , v {\boldsymbol{v}} , and w {\boldsymbol{w}} that define continuous and compact tridiagonal operators T u , v , w {T}_{u,v,w} acting on the weighted sequence space l β 2 {l}_{\beta }^{2} were characterized. Additionally, the essential norm of this operator, and as an important consequence of our results, the essential norm of multiplication operator M u {M}_{u} acting on l β 2 {l}_{\beta }^{2} spaces was calculated.
