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Abstract: Abstract In this paper, the author introduces Triebel–Lizorkin spaces with general smoothness. We present the \(\varphi\) -transform characterization of these spaces in the sense of Frazier and Jawerth and we prove their Sobolev embeddings. Also, we establish the smooth atomic and molecular decomposition of these function spaces. To do these we need a generalization of some maximal inequality to the case of general weights. PubDate: 2022-11-15

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Abstract: Abstract A closed densely defined operator T on a Hilbert space \({\mathcal {H}}\) is called M-hyponormal if \({\mathcal {D}}(T) \subset {\mathcal {D}}(T^{*})\) and there exists \(M>0\) for which \(\Vert (T-zI)^{*}x \Vert \le M \Vert (T-zI)x \Vert \) for all \(z \in {\mathbb {C}}\) and \(x\in {\mathcal {D}}(T)\) . In this paper, we prove that if \(A:{\mathcal {H}}\rightarrow {\mathcal {K}}\) is a bounded linear operator, such that \(AB^*\subseteq TA\) , where B is a closed subnormal (resp. a closed M-hyponormal) on \({\mathcal {H}}\) , T is a closed M-hyponormal (resp. a closed subnormal) on a Hilbert space \({\mathcal {K}}\) , then (i) \( AB\subseteq T^*A\) (ii) \({\overline{\text{ ran }\,A^{*}}}\) reduces B to the normal operator \( B\vert _{{\overline{\text{ ran }\,A^{*}}}}\) and (iii) \({\overline{\text{ ran }\,A}}\) reduces T to the normal operator \( T\vert _{{\overline{\text{ ran }\,A}}}\) . PubDate: 2022-11-14

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Abstract: Abstract The Hermite–Hadamard inequalities attract many researchers and are useful theoretical point of view as well as in practical purposes. Some properties, extensions, refinements, reverses and applications of these inequalities have been carried out recently. In this paper, we investigate some refinements and reverses for the Hermite–Hadamard inequalities when the integrand map is operator convex (resp. operator concave) in multiple operator arguments. As application, some inequalities for multivariate operator means are provided. PubDate: 2022-11-07

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Abstract: Abstract In this article, we focus on the continuity as well as boundedness properties of fractional Hankel-like transform and pseudo-differential operator associated with a fractional Hankel-like transform on some suitably designed Gelfand–Shilov spaces of type \({\mathcal {S}}\) . Moreover, we have further investigated on certain class of ultradifferentiable function spaces for the above integral transform and operator. PubDate: 2022-10-31

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Abstract: Abstract This paper concerns existence of positive solutions for a second-order boundary value problem associated with a \(\phi \) -Laplacian operator and the four point boundary conditions and posed on the bounded interval. Existence results are obtained by an adapted version of the Krasnosel’skii’s fixed point theorem of cone expansion and compression. Some examples illustrate our existence results. PubDate: 2022-10-23

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Abstract: Abstract We prove that a Kreiss bounded \(C_0\) semigroup \((T_t)_{t \ge 0}\) on a Hilbert space has asymptotics \(\left\ T_t\right\ = {\mathcal{O}}\big (t/\sqrt{\log (t)}\big ).\) Then, we give an application to perturbed wave equation. PubDate: 2022-10-06

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Abstract: Abstract This paper defines two definitions for Sobolev-type spaces using fractional Hankel–Clifford transform pairs. We constructed an example in each space and obtained its solution. We also proved that these spaces are complete. Further, we derived the boundedness properties of the product and convolution of these spaces. Furthermore, we provided some applications of fractional Hankel–Clifford transform pairs to solve analytically generalized homogeneous partial differential equations with variable coefficients. Finally, we illustrated the solution surface profiles for generalized homogeneous partial differential equations with variable coefficients in 3D view to show the analytical solution more elegantly. PubDate: 2022-10-05

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Abstract: Abstract In this paper, a brief study is made on the Legendre function related to the Lebedev–Skalskaya transform (LSL-transform) pairs. Some preliminary useful results are obtained. The translation and convolution operators are introduced and obtained their estimates on Lebesgue space. Further, continuity mapping of LSL-transforms, the translation and convolution operators are discussed over the function spaces \({\mathfrak {H}}^{m,n}\) and \({\mathfrak {G}}^{k,n}\) . Moreover, pseudo-differential operators involving LSL-transforms are defined and studied their continuity property over the spaces \({\mathfrak {H}}^{m,n}\) and \({\mathfrak {G}}^{k,n}\) . An integral representation of pseudo-differential operators are also given. PubDate: 2022-10-03

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Abstract: Abstract In this paper, we characterize Carleson measures on the Dirichlet type space \(\mathcal D_{\alpha }^{p}\) for different values of p, q and \(\alpha\) and in terms of axially symmetric completion of a pseudohyperbolic disc. PubDate: 2022-10-01

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Abstract: Abstract The w*-rigged modules over dual operator algebras were introduced by Blecher and Kashyap as a generalization of W*-modules. In this paper, we introduce two new types of Morita equivalence between right w*-rigged modules over unital dual operator algebras and we examine whether these notions imply stable isomorphism between the corresponding modules. Furthermore, we investigate them in detail for the class of right w*-rigged modules over nest algebras, a class which was characterized by G.K. Eleftherakis. PubDate: 2022-09-28

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Abstract: Abstract A broader class of Hardy spaces and Lebesgue spaces have been introduced recently on the unit circle by considering continuous \(\Vert \cdot \Vert _1\) -dominating normalized gauge norms instead of the classical norms on measurable functions. A Beurling type result has been proved for the operator of multiplication by the coordinate function. In this paper, we generalize the above Beurling type result to the context of multiplication by a finite Blaschke factor B(z) and also derive the common invariant subspaces of \(B^2(z)\) and \(B^3(z).\) These results lead to a factorization result for all functions in the Hardy space equipped with a continuous rotationally symmetric norm. PubDate: 2022-09-27

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Abstract: Abstract In this paper, we investigate a problem of Halmos on various generalizations of the numerical range. We generalize a finite-dimensional result of Gau, Li and Wu, by showing that for \(k\in {\mathbb {N}},\) the closure of the rank-k numerical range of a contraction T acting on a separable Hilbert space \({\mathcal{H}}\) is the intersection of the closure of the rank-k numerical ranges of all unitary dilations of T to \({\mathcal{H}}\oplus {\mathcal{H}}.\) The same is true for \(k=\infty\) provided the rank- \(\infty\) numerical range of T is non-empty. We also show that when both defect indices of a contraction are equal and finite ( \(=N\) ), one may restrict the intersection to a smaller family consisting of all unitary N-dilations. We also investigate this problem in the matricial range and the C-numerical range. We obtain a few interesting results and conclude the answers in negative. PubDate: 2022-09-23

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Abstract: Abstract We show several operator inequalities concerning Young’s inequality by using relative operator entropies and operator valued \(\alpha \) -divergence. One of them is partially a refinement of Young’s inequality. PubDate: 2022-09-21 DOI: 10.1007/s43036-022-00220-2

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Abstract: Abstract We study the operator \(\mathcal {A}\) of multiplication by an independent variable in a matrix Sobolev space \(W^2(M)\) . In the cases of finite measures on [a, b] with \((2\times 2)\) and \((3\times 3)\) real continuous matrix weights of full rank it is shown that the operator \(\mathcal {A}\) is symmetrizable. Namely, there exist two symmetric operators \(\mathcal {B}\) and \(\mathcal {C}\) in a larger space such that \(\mathcal {A} f = \mathcal {C} \mathcal {B}^{-1} f\) , \(f\in D(\mathcal {A})\) . As a corollary, we obtain some new orthogonality conditions for the associated Sobolev orthogonal polynomials. These conditions involve two symmetric operators in an indefinite metric space. PubDate: 2022-09-19 DOI: 10.1007/s43036-022-00221-1

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Abstract: Abstract In the present work, using the power series method, we obtain various Korovkin-type approximation theorems for linear operators defined on derivatives of functions. We also explain that our theorem makes more sense with a striking example. We study the quantitative estimates of linear operators. In the final section, we summarize our new results. PubDate: 2022-09-17 DOI: 10.1007/s43036-022-00218-w

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Abstract: Abstract In this study of non-commutative convexity in the context of operator spaces, we explore extremal nature of embedding of an operator space in its injective envelope. We introduce the notions of bi-extremity of completely contractive (CC) maps on operator spaces and of prime TROs generated by them in the context of Hilbert \(C^*\) -bimodules. It is shown that if the generated TRO of an operator space is prime, then the CC embedding of the operator space in its injective envelope is bi-extreme. PubDate: 2022-09-04 DOI: 10.1007/s43036-022-00214-0

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Abstract: Abstract We study the solvability and stability of the Tikhonov regularized equation of the Fredholm integral equations of the first kind using Legendre spectral multi-projection method. We discuss the theoretical analysis of this method with the convergence analysis under a priori parameter choice strategy for the Tikhonov regularization using Legendre polynomial basis functions. We obtain the optimal convergence rates in infinity norm. Next, we discuss the Arcangeli’s discrepancy principle to find a suitable regularization parameter and obtain optimal order of convergence in infinity norm. We provide numerical examples to illustrate our theoretical results. PubDate: 2022-08-30 DOI: 10.1007/s43036-022-00215-z

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Abstract: Abstract This article presents some refinements and generalizations of existing numerical radius inequalities for Hilbert space operators. Further, several upper bounds for numerical radius with the Cartesian decomposition of an operator are provided. Also, we prove various upper bounds for the numerical radius of diagonal and off-diagonal operator matrices. Additionally, we establish certain numerical radius inequalities for operators using quadratic weighted operator geometric mean. PubDate: 2022-08-29 DOI: 10.1007/s43036-022-00216-y

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Abstract: Abstract In this paper, we study the notion of super-recurrence for a strongly continuous semigroup of operators. We establish some results for super-recurrent \(C_{0}\) -semigroups. As an application, we study the super-recurrence of the translation \(C_{0}\) -semigroup. PubDate: 2022-08-26 DOI: 10.1007/s43036-022-00213-1

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Abstract: Abstract In this paper, we obtain two Lichnerowicz type formulas for the operators \(\sqrt{-1}\widehat{c}(V)(d+\delta )\) and \(-\sqrt{-1}(d+\delta )\widehat{c}(V)\) . And we give the proof of Kastler–Kalau–Walze type theorems for the operators \(\sqrt{-1}\widehat{c}(V)(d+\delta )\) and \(-\sqrt{-1}(d+\delta )\widehat{c}(V)\) on 3,4-dimensional oriented compact manifolds with (resp.without) boundary. PubDate: 2022-08-01 DOI: 10.1007/s43036-022-00211-3