We introduce a generalized framework for the Kaczmarz algorithm adapted to bounded linear operators, defining the notion of operator-weighted effective and almost effective sequences. In contrast to the classical Kaczmarz algorithm, our approach enables the reconstruction of elements from the range of such operators without the necessary condition of a total sequence in the ambient space. Operator-weighted effective sequences and their variations are characterized in terms of the operator-weighted frame properties of various associated sequences. We also prove that operator-weighted special Bessel sequences can be used to construct operator-weighted effective and almost effective sequences.
We study the Fredholm index theory for commuting operator tuples, a central topic in multivariate operator theory. Building upon known results for pairs of commuting isometries, we investigate the relationship between the Fredholm indices of a triple of commuting isometries V = (V_1, V_2, V_3) on a separable Hilbert space ℋ and those of its associated fringe operator pair F = (F_1, F_2) . Using exterior algebra and Koszul complex theory, we construct the relevant complex structures for both operator tuples and clarify the definitions and methods for computing their Fredholm indices. By establishing explicit isomorphisms, we show that the homology groups satisfy H_i(F) ≅ H_i+1(V) for i = 0, 1, 2 , providing an equivalent criterion for their Fredholmness. As a key result, we prove that the Fredholm indices of the fringe operator pair and the triple operator satisfy ind(F) = -ind(V) .
In this paper, we introduce multilinear fractional bounded oscillation operators on abstract measure Lebesgue spaces with the ball–basis structure. The weighted estimates for the multilinear fractional sparse operators are established. Furthermore, the weighted estimates for these multilinear fractional bounded oscillation operators are also obtained. Moreover, the exploration of multilinear fractional bounded oscillation operators offers a view of several classical operators in harmonic analysis, including the multilinear fractional Hardy–Littlewood maximal operators, multilinear Carleson–type operators and the Riesz potential. As applications, we obtain sharp weighted estimates for these operators.
We propose, for the first time, a Caputo-type theory of fractional powers of the complex d-bar derivative. Our investigation of these operators includes their composition properties, relations with fractional d-bar derivatives of Riemann–Liouville type, and a proof that, in this setting, fractional polyanalyticity is equivalent to classical polyanalyticity, due to the nature of Caputo-type operators.
