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In this paper we consider the following problem: let Xk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by ${(e_{(k,j)}^*)_j}$ , for $k\\in\\N$ , let $Z=\\ell^\\infty(X_k:k\\kin\\N)$ be their l∞-sum, and let $T:Z\\to Z$ be a bounded linear operator with a large diagonal, i.e., $$\\begin{align*}\\inf_{k,j} \\big|e^*_{(k,j)}(T(e_{(k,j)})\\big|>0.\\end{align*}$$ Under which condition does the identity on Z factor through T? The purpose of this paper is to formulate general conditions for which the answer is positive.

We prove that a generic function from ⋂ p < 1 H p has unbounded Taylor coefficients, this applies also to the Taylor coefficients of its derivatives. Results of similar nature are valid when the space ⋂ p < 1 H p is replaced by H p ( 0 < p < 1 ) and by localized versions of such spaces. Moreover, we prove that a generic function from A ( D ) has Taylor coefficients outside of ℓ 1 , this applies also to the Taylor coefficients of its derivatives. Lastly, we prove that a generic function from ⋂ p < 1 h p has a harmonic conjugate that does not belong to any h q ( q > 0 ) .

This paper considers paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space H 2 . The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly S ∗ -invariant subspace of H 2 , is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.

Let α > - 1 and assume that f is α -harmonic mapping defined in the unit disk that belongs to the Hardy class h p with p ⩾ 1 . We obtain some sharp estimates of the type | f ( z ) | ≤ g ( | r | ) ‖ f ∗ ‖ p and | D f ( z ) | ≤ h ( | r | ) ‖ f ∗ ‖ p . We also prove a Schwarz type lemma for the class of α -harmonic mappings of the unit disk onto itself fixing the origin.

The invariant subspace problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota) the ISP can be solved by proving that every minimal invariant subspace of a universal operator is one dimensional. In this work, we obtain conditions for T φ ∗ | M to have a non-trivial subspace where M ⊂ H 2 ( D 2 ) is an invariant subspace of the Toeplitz operator T φ ∗ on the Hardy space over the bidisk H 2 ( D 2 ) induced by the symbol φ ∈ H ∞ ( D ) . We then use this fact to obtain sufficient conditions for the ISP to be true.

This paper presents a fairly general version of the well-known Gleason–Kahane– Z ˙ elazko (GKZ) theorem in the spirit of a GKZ type theorem obtained recently by Mashreghi and Ransford for Hardy spaces. In effect, we characterize a class of linear functionals as point evaluations on the vector space of all complex polynomials P . We do not make any topological assumptions on P . We then apply this characterization to present a version of the GKZ theorem for a vast class of topological spaces of complex-valued functions including the Hardy, Bergman, Dirichlet, and many more well-known function spaces. We obtain this result under the assumption of continuity of the linear functional, which we show, with the help of an example, to be a necessary condition for the desired conclusion. Lastly, we use the GKZ theorem for polynomials to obtain a version of the GKZ theorem for strictly cyclic weighted Hardy spaces.

We establish the global regularity of multilinear Fourier integral operators that are associated to nonlinear wave equations on products of$L^p$spaces by proving endpoint boundedness on suitable product spaces containing combinations of the local Hardy space, the local BMO and the$L^2$spaces.

Inspired by Xiao’s work about the Hankel measures for the weighted Bergman spaces, in this paper, if β > 0 and the measure μ is a complex Borel measure on the unit disk D , we define the Hankel type operator K μ , β by K μ , β : f ⟼ ∫ D ( 1 - w z ) - ( β ) f ( w ) d μ ( w ) . The operator itself has been widely studied when μ is a positive Borel measure supported on the interval [0, 1). We study the boundedness of K μ , 1 acting on Hardy spaces and the boundedness of K μ , α , α > 1 acting on weighted Bergman spaces. Then we raise and answer some questions about the boundedness of those operators. Also, we find some special measures μ ′ s such that s -Hankel measure is equal to s -Carleson measure.

In this article, we will give an affirmative answer to Gilbert’s conjecture on Hardy spaces of Clifford analytic functions in upper half-space of R 8 . It is based on an explicit construction of Clifford algebra C l 8 and Spinor space R 8 by octonion algebra. Furthermore, it provides an associative approach to the theory of octonionic analytic functions. Additionally, certain classical results about octonionic analytic functions have been reformulated, and a related subject has been treated in Octonionic Hardy space in upper half-space.