Dual Toeplitz Operators on the Sphere Via Spherical Isometries

The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results. (1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules. (2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators. (3) Theorem III gives several characterizations of an analytic Toeplitz operator. (4) Theorem IV characterizes asymptotic Toeplitz operators. (5) Theorem V is a commutant lifting theorem. (6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

Toeplitz and asymptotic Toeplitz operators on [formula omitted

We study the commuting problem for Toeplitz operators on the Hardy space of the polydisk. We consider two symbols where one is an arbitrarily bounded symbol and the other is a bounded symbol which has the form of a sum of an analytic function and a co-analytic function. We then give a necessary and sufficient condition on the symbols for which the corresponding Toeplitz operators commute each other.

We study the commuting problem for Toeplitz operators on the harmonic Bergman space of the unit disk. We show that an analytic Toeplitz operator and a co-analytic Toeplitz operator with certain noncyclicity hypothesis can commute only when one of their symbols is constant. We also obtain analogous results for semi-commutants.

Block dual Toeplitz operators on the orthogonal complement of the Dirichlet space

In this paper one considers some general theorems of the theory of Hankel and Toeplitz operators in spaces of analytic functions. Under natural restrictions on the spaces X, it is shown that the symbols of the Toeplitz operators, acting in X, are bounded. One describes completely the symbols of the Hankel and Toeplitz operators, acting from Hp into ¯Hq (into Hq) for 0 < p, q < ∞.

We consider Toeplitz operators $T_f^{\lambda}$ with symbol $f$ acting on the standard weighted Bergman spaces over a bounded symmetric domain $\Omega\subset \mathbb{C}^n$. Here $\lambda > genus-1$ is the weight parameter. The classical asymptotic semi-commutator relation $\lim_{\lambda \rightarrow \infty} \big{\|}T_f^{\lambda} T_g^{\lambda} -T_{fg}^{\lambda} \big{\|}=0$ with $f,g \in C(\overline{\mathbb{B}^n})$, where $\Omega=\mathbb{B}^n$ denotes the complex unit ball, is extended to larger classes of bounded and unbounded operator symbol-functions and to more general domains. We deal with operator symbols that generically are neither continuous inside $\Omega$ (Section 4) nor admit a continuous extension to the boundary (Section 3 and 4). Let $\beta$ denote the Bergman metric distance function on $\Omega$. We prove that the semi-commutator relation remains true for $f$ and $g$ in the space ${\rm UC}(\Omega)$ of all $\beta$-uniformly continuous functions on $\Omega$. Note that this space contains also unbounded functions. In case of the complex unit ball $\Omega=\mathbb{B}^n \subset \mathbb{C}^n$ we show that the semi-commutator relation holds true for bounded symbols in ${\rm VMO}(\mathbb{B}^n)$, where the vanishing oscillation inside $\mathbb{B}^n$ is measured with respect to $\beta$. At the same time the semi-commutator relation fails for generic bounded measurable symbols. We construct a corresponding counterexample using oscillating symbols that are continuous outside of a single point in $\Omega$.

Tense operators for MV-algebras were introduced by Diaconescu and Georgescu. Based on their denition Chajda and Kolařik presented the denition of tense operators for lattice effect algebras. Chajda and Paseka tackled the problem of axiomatizing tense operators on an effect algebra by introducing the notion of a partial dynamic effect algebra. They also gave representation theorems for dynamic effect algebras. We continue to extend their work for partial S-dynamic effect algebras i.e. in the case when tense operators satisfy required conditions also for the dual effect algebraic operation . We show that whenever tense operators are total our stronger notion coincides with their denition. We give also a representation theorem for partial S-dynamic effect algebras and its version for strict dynamic effect algebras.

For each automorphic composition operator C ϕ acting on either the Hardy or Bergman Hilbert space of the unit ball in , we show that is a Toeplitz operator, that is the inverse of a Toeplitz operator, and that the selfcommutator is essentially a Toeplitz operator. We then extract spectral and norm information from these identifications, focusing on selfcommutators. For example, in the setting of the Hardy space of the unit disk, we obtain complete descriptions of the spectrum, essential spectrum, and point spectrum for selfcommutators of automorphic composition operators, which reveal that the spectrum and essential spectrum coincide.

In this paper, we investigate the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a class of operators that we call radial operators, an oscillation criterion and diagonal are sufficient conditions under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit sphere. We further study a special class of radial operators, i.e., Toeplitz operators with a radial L1(Bn) symbol.

In this article, we introduce a new class of conjugations in the scalar-valued Hardy space and provide a characterization of a complex symmetric Toeplitz operator with respect to these newly introduced conjugations in various cases. Moreover, we obtain a characterization of a complex symmetric block Toeplitz operator on the vector-valued Hardy space with respect to certain conjugations introduced in [Câmara MC, Kliś-Garlicka K, Ptak M. Complex symmetric completions of partial operator matrices. Linear and Multilinear Algebra. 2019; DOI: 10.1080/03081087.2019.1631246], [Kang D, Ko E, Lee JE. Remarks on complex symmetric Toeplitz operators. Linear Multilinear Algebra. 2020; DOI: 10.1080/03081087.2020.1842847], [Ko E, Lee JE. Remark on complex symmetric operator matrices. Linear Multilinear Algebra. 2019;67(6):1198–1216].

We characterize the topologizability and power boundedness of convolution and dual convolution operators on power series spaces. We determine necessary conditions for a Toeplitz operator to be m-topologizable, and power bounded on $$\Lambda _{1}(n)$$ Λ 1 ( n ) and $$\Lambda _{\infty }(n)$$ Λ ∞ ( n ) , and consequently on $$H(\mathbb {C})$$ H ( C ) and $$H(\mathbb {D})$$ H ( D ) .

On Schatten-class perturbations of Toeplitz operators

Toeplitz Operators and Hankel Operators on the Hardy Space of the Unit Sphere

Let $$\Gamma $$ be a rectifiable Jordan curve, let X and Y be two reflexive Banach function spaces over $$\Gamma $$ such that the Cauchy singular integral operator S is bounded on each of them, and let M(X, Y) denote the space of pointwise multipliers from X to Y. Consider the Riesz projection $$P=(I+S)/2$$ , the corresponding Hardy type subspaces PX and PY, and the Toeplitz operator $$T(a):PX\rightarrow PY$$ defined by $$T(a)f=P(af)$$ for a symbol $$a\in M(X,Y)$$ . We show that if $$X\hookrightarrow Y$$ and $$a\in M(X,Y)\setminus \{0\}$$ , then $$T(a)\in \mathcal {L}(PX,PY)$$ has a trivial kernel in PX or a dense image in PY. In particular, if $$1<q\le p<\infty $$ , $$1/r=1/q-1/p$$ , and $$a\in L^{r}\equiv M(L^p,L^q)$$ is a nonzero function, then the Toeplitz operator T(a), acting from the Hardy space $$H^p$$ to the Hardy space $$H^q$$ , has a trivial kernel in $$H^p$$ or a dense image in $$H^q$$ .