Block-Toeplitz Operators On the Hardy Space Induced by a Tracial Unital Banach $$*$$-Probability Space

We continue Curto–Hwang–Lee’s study of the connection between hyponormality and subnormality for block Toeplitz operators acting on the vector-valued Hardy space of the unit circle. Curto–Hwang–Lee’s work focuses primarily on block Toeplitz operators with rational symbols. By studying the greatest common divisor of matrix-valued inner functions and the “weak” commutativity of matrix-valued inner functions, we extend Curto–Hwang–Lee’s result to block Toeplitz operators with symbols of bounded type. More precisely, we prove that if Ψ,Ψ∗ are matrix-valued functions of bounded type and the inner part of the Douglas–Shapiro–Shields factorization of Ψ is a scalar inner function, then every hyponormal Toeplitz operator TΨ whose square is also hyponormal must be either normal or analytic.

In this paper, we give an alternative characterization of complex symmetry of Toeplitz operators and block Toeplitz operators. In particular, we prove that a block Toeplitz operator is complex symmetric with the conjugation on the vector-valued Hardy space if and only if , where denotes the multiplication operator on with symbol Φ. As some applications, we provide examples of normal (resp., non-normal) complex symmetric block Toeplitz operators.

In this paper we characterize when the product of two block Toeplitz operators is a compact perturbation of a block Toeplitz operator on the Hardy space of the open unit disk. Necessary and sufficient conditions are given for the commutator of two block Toeplitz operators to be compact.

Hyponormality and subnormality of block Toeplitz operators

This paper is mainly devoted to establishing an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces. More precisely, let (Ω,F, ℙ) be a probability space and p(·): Ω →(0,∞) be a F-measurable function such that $$0 < {\inf _{x \in \Omega }}p(x) \leqslant {\sup _{x \in \Omega }}p(x) < \infty $$ . It is proved that a predictable martingale Hardy space P p (·) has an atomic decomposition by some key observations and new techniques. As an application, we obtain the boundedness of fractional integrals on the predictable martingale Hardy space with variable exponents when the stochastic basis is regular.

In this paper we investigate the boundedness of fractional integral operators on predictable martingale Hardy spaces with variable exponents defined on a probability space. More precisely, let f = (fn)n≥0 be a martingale on probability space (Ω,F, ℙ), and let Iαf, α &gt; 0 be the fractional integral operator associated with f. Under some reasonable assumptions, it is proved that Iαf is bounded on martingale Hardy spaces with variable exponents. Our method is an extension of atomic decomposition theorem to predicable martingale Hardy spaces of variable exponents.

Given a probability space $(Ω,Σ,μ)$, the Hardy space $\mathrm{H}_1(Ω)$ which is associated to the martingale square function does not admit a classical atomic decomposition when the underlying filtration is not regular. In this paper we construct a decomposition of $\mathrm{H}_1(Ω)$ into "atomic blocks"${}$ in the spirit of Tolsa, which we will introduce for martingales. We provide three proofs of this result. Only the first one also applies to noncommutative martingales, the main target of this paper. The other proofs emphasize alternative approaches for commutative martingales. One might be well-known to experts, using a weaker notion of atom and approximation by atomic filtrations. The last one adapts Tolsa's argument replacing medians by conditional medians.

New martingale inequalities and applications to Fourier analysis

In this paper, we introduce Hardy spaces with variable exponents defined on a probability space and develop the martingale theory of variable Hardy spaces. We prove the weak-type and strong-type inequalities on Doob’s maximal operator, and we get a (1,p(⋅),∞)-atomic decomposition for Hardy martingale spaces associated with conditional square functions. As applications, we obtain a dual theorem and the John–Nirenberg inequalities in the frame of variable exponents. The key ingredient is that we find a condition with a probabilistic characterization of p(⋅) to replace the so-called log-Hölder continuity condition in Rn.

Let (Ω,F,P) be a probability space, and let φ:Ω×[0,∞)→[0,∞) be a Musielak–Orlicz function. In this article, we establish the atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces WHφs(Ω), WHφM(Ω), WHφS(Ω), WPφ(Ω), and WQφ(Ω). We then use these atomic characterizations to obtain the boundedness of σ-sublinear operators from weak martingale Musielak–Orlicz Hardy spaces to weak Musielak–Orlicz spaces, as well as some martingale inequalities which further clarify the relationships among these weak martingale Musielak–Orlicz Hardy spaces. All these results improve and generalize the corresponding results on weak martingale Orlicz–Hardy spaces. Moreover, we improve all the known results on weak martingale Musielak–Orlicz Hardy spaces. In particular, both the boundedness of σ-sublinear operators and the martingale inequalities, for weak weighted martingale Hardy spaces as well as for weak weighted martingale Orlicz–Hardy spaces, are new.

Let $$(\Omega ,{\mathcal {F}},{\mathbb {P}})$$ be a probability space, $$\varphi :\ \Omega \times [0,\infty )\rightarrow [0,\infty )$$ a Musielak–Orlicz function, and $$q\in (0,\infty ]$$ . In this article, the authors introduce five martingale Musielak–Orlicz–Lorentz Hardy spaces and prove that these new spaces have some important features such as atomic characterizations, the boundedness of $$\sigma $$ -sublinear operators, and martingale inequalities. This new scale of martingale Hardy spaces requires the introduction of the Musielak–Orlicz–Lorentz space $$L^{\varphi ,q}(\Omega )$$ . In particular, the authors show that this Lorentz type space has some fundamental properties including the completeness, the convergence, real interpolations, and the Fefferman–Stein vector-valued inequality for the Doob maximal operator. As applications, the authors prove that the maximal Fejer operator is bounded from the martingale Musielak–Orlicz–Lorentz Hardy space $$H_{\varphi ,q}[0,1)$$ to $$L^{\varphi ,q}[0,1)$$ , which further implies some convergence results of the Fejer means. Moreover, all the above results are new even for Musielak–Orlicz functions with particular structure such as weight, weight Orlicz, and double-phase growth. The main approach used in this article can be viewed as a combination of the stopping time argument in probability theory and the real-variable technique of function spaces in harmonic analysis.

Let (Ω,F, P) be a probability space and {F n}n≥0 a regular increasing sequence of sub-σ-fields ofF. LetH 1(Ω) be the usual Hardy space ofF n-martingales. We show that the couple (H 1(Ω),L ∞(Ω)) is a partial retract of (L 1(Ω),L ∞(Ω)). It is also proved that (L p(Ω),BMO(Ω)) is a partial retract of (L p(Ω),L ∞(Ω)) for all 1<p<∞.

We introduce grand Hardy spaces defined on a probability space. Analogous to the classical theory, we prove Doob’s maximal inequality and obtain atomic characterization of grand Hardy martingale spaces. Finally, we investigate the John–Nirenberg theorem in the frame of grand Hardy spaces.

Real interpolation for variable martingale Hardy spaces

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].