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Abstract: Abstract The invertibility of Toeplitz plus Hankel operators \(T(\mathcal {A})+H(\mathcal {B})\) , \(\mathcal {A},\mathcal {B}\in L^\infty _{d\times d}(\mathbb {T})\) acting on vector Hardy spaces \(H^p_d(\mathbb {T})\) , \(1<p<\infty \) , is studied. Assuming that the generating matrix functions \(\mathcal {A}\) and \(\mathcal {B}\) satisfy the equation $$\begin{aligned} \mathcal {B}^{-1} \mathcal {A}= \widetilde{\mathcal {A}}^{-1}\widetilde{\mathcal {B}}, \end{aligned}$$ where \(\widetilde{\mathcal {A}}(t):=\mathcal {A}(1/t)\) , \(\widetilde{\mathcal {B}}(t):=\mathcal {B}(1/t)\) , \(t\in \mathbb {T}\) , we establish sufficient conditions for the one-sided invertibility and invertibility of the operators mentioned and construct the corresponding inverses. If \(d=1\) , the above equation reduces to the known matching condition, widely used in the study of Toeplitz plus Hankel operators with scalar generating functions. PubDate: 2024-08-03 DOI: 10.1007/s43036-024-00373-2
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Abstract: Abstract We prove some splitting results for bilinear summing operators and as a consequence Pietsch type composition results. Some examples are given. PubDate: 2024-07-29 DOI: 10.1007/s43036-024-00372-3
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Abstract: Abstract In 1996, X. Li and D. Yang found the best possible range of index \(\alpha \) for the boundedness of some sublinear operators on Herz spaces \({\dot{K}}_q^{\alpha , p}({{\mathbb {R}}}^n)\) or \(K_q^{\alpha , p}({{\mathbb {R}}}^n)\) , under a certain size condition. Also, in 1994 and 1995, S. Lu and F. Soria showed that concerning the boundedness of above sublinear operator T on \({\dot{K}}_q^{\alpha , p}({{\mathbb {R}}}^n)\) or \(K_q^{\alpha , p}({{\mathbb {R}}}^n)\) with critical index of \(\alpha \) , T is bounded on the power-weighted Herz spaces \({\dot{K}}_q^{\alpha , p}(w)({{\mathbb {R}}}^n)\) or \(K_q^{\alpha , p}(w)({{\mathbb {R}}}^n)\) . In this paper, we will prove that for the two-power-weighted Herz spaces \({\dot{K}}_{q_1}^{\alpha , p}(w_1,w_2)({{\mathbb {R}}}^n)\) or \(K_{q_2}^{\alpha , p}(w_1,w_2)({{\mathbb {R}}}^n)\) with indices beyond critical index of \(\alpha \) , the above T is bounded on them. Further, we will extend this result to a sublinear operator satisfying another size condition and a pair of Herz spaces \(K_q^{\alpha , p}(w_{\beta _1},w_{\beta _2})({{\mathbb {R}}}^n)\) and \(K_q^{\alpha , p}(w_{\gamma _1},w_{\gamma _2})({{\mathbb {R}}}^n)\) . Moreover, we will also show the result of weak version of the above boundedness. PubDate: 2024-07-27 DOI: 10.1007/s43036-024-00368-z
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Abstract: Abstract In this paper, we prove several interpolating inequalities for unitarily invariant norms of matrices. Using the log-convexity of certain functions, enables us to obtain refinements of recent norm inequalities. Generalizations of some well-known norm inequalities are also given. PubDate: 2024-07-27 DOI: 10.1007/s43036-024-00371-4
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Abstract: Abstract In this paper, we study the commutators of the Hardy operators on the Heisenberg group. We get some sufficient and necessary conditions for the compactness of the commutators of the Hardy operators on the Heisenberg group. PubDate: 2024-07-25 DOI: 10.1007/s43036-024-00369-y
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Abstract: Abstract The \(\exp _q(z)\) function is the standard q-analogue of the exponential. Since not much is known about this function, our aim is to give a contribution to the knowledge on \(\exp _q\) . After proving some simpler but new relations for it, we make a complete description of the inverse map of \(\exp _q(z)\) , including its branch structure and Riemann surface. PubDate: 2024-07-23 DOI: 10.1007/s43036-024-00367-0
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Abstract: Abstract We prove an analogue of the Brown–Halmos theorem for discrete Wiener–Hopf operators acting on separable rearrangement-invariant Banach sequence spaces. PubDate: 2024-07-17 DOI: 10.1007/s43036-024-00370-5
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Abstract: Abstract In this paper, we characterize the boundedness of weighted composition operators, induced by measurable transformations and complex-valued measurable functions, on variable exponent Lebesgue spaces. We also derive conditions for these operators to be compact or injective or have closed range. In addition, we investigate some relations between these operators and multiplication operators. PubDate: 2024-07-13 DOI: 10.1007/s43036-024-00366-1
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Abstract: Abstract In the present paper we introduce a method for generating ideals of linear and multilinear operators from what we call generalized left and right operator ideals, that we discuss with p-th power factorable, p-convex and q-concave operators. Then we combine this method with the Factorization Ideal method, that construct multilinear operators, in order to introduce the ideal of multilinear \({\mathcal {F}}_{\vec {p},\vec {q}}\) -factorable operators as an example of an ideal generated by means of our method. Finally, we investigate its relation with multilinear summing operators. PubDate: 2024-07-06 DOI: 10.1007/s43036-024-00365-2
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Abstract: Abstract Let \(\mathfrak {M}\) be a von Neumann algebra. For a nonzero positive element \(A\in \mathfrak {M}\) , let P denote the orthogonal projection on the norm closure of the range of A and let \(\sigma _A(T) \) denote the A-spectrum of any \(T\in \mathfrak {M}^A\) . In this paper, we show that \(\sigma _A(T)\) is a non empty compact subset of \(\mathbb {C}\) and that \(\sigma (PTP, P\mathfrak {M}P)\subseteq \sigma _A(T)\) for any \(T\in \mathfrak {M}^A\) where \(\sigma (PTP, P\mathfrak {M}P)\) is the spectrum of PTP in \(P\mathfrak {M}P\) . Sufficient conditions for the equality \(\sigma _A(T)=\sigma (PTP, P\mathfrak {M}P)\) to be true are also presented. Moreover, we show that \(\sigma _A(T)\) is finite for any \(T\in \mathfrak {M}^A\) if and only if A is in the socle of \(\mathfrak {M}\) . Furthermore, we consider the relationship between elements S and \(T\in \mathfrak {M}^A\) that satisfy the condition \(\sigma _A(SX)=\sigma _A(TX)\) for all \(X\in \mathfrak {M}^A\) . Finally, a Gleason–Kahane–Żelazko’s theorem for the A-spectrum is derived. PubDate: 2024-07-04 DOI: 10.1007/s43036-024-00362-5
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Abstract: Abstract We say that a \(C^*\) -algebra \({\mathcal {A}}\) satisfies the similarity property ((SP)) if every bounded homomorphism \(u: {\mathcal {A}}\rightarrow {\mathcal {B}}(H),\) where H is a Hilbert space, is similar to a \(*\) -homomorphism and that a von Neumann algebra \({\mathcal {M}}\) satisfies the weak similarity property ((WSP)) if every \(\textrm{w}^*\) -continuous, unital and bounded homomorphism \(\pi : {\mathcal {M}}\rightarrow {\mathcal {B}}(H),\) where H is a Hilbert space, is similar to a \(*\) -homomorphism. The similarity problem is known to be equivalent to the question of whether every von Neumann algebra is hyperreflexive. We improve on that by introducing the following hypothesis (EP): Every separably acting von Neumann algebra with a cyclic vector is hyperreflexive. We prove that under (EP), every separably acting von Neumann algebra satisfies (WSP) and we pass from the case of separably acting von Neumann algebras to all \(C^*\) -algebras. PubDate: 2024-07-02 DOI: 10.1007/s43036-024-00363-4
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Abstract: Abstract Pólya-type functions are of special importance in probability and harmonic analysis. We introduce and study their multidimensional extensions. PubDate: 2024-06-21 DOI: 10.1007/s43036-024-00361-6
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Abstract: Abstract Let A, B be any two positive definite \(n\times n\) matrices and Y be any \(n\times n\) matrix. The matrices \(M_Y(A,B)=\left[ \begin{array}{cc} A &{} A^{\frac{1}{2}}YB^{\frac{1}{2}} \\ B^{\frac{1}{2}}Y^{\star }A^{\frac{1}{2}} &{} B \end{array}\right] \) for Y to be contractive, expansive or unitary matrix, are in fact arising from matrix/operator means. We aim to establish the signatures of the eigenvalues of the sum of two matrices of the type \(M_Y(A,B).\) We characterise any \(n\times n\) matrix A through its dilations given by \({\mathcal {P}}_3(A)=\begin{bmatrix} O &{} A &{} A^2\\ A^* &{} O &{} A\\ {A^*}^2 &{} A^* &{} O \end{bmatrix}\) and \({\mathcal {M}}_3(A)=\begin{bmatrix} I &{} A &{} A^2\\ A^* &{} I &{} A\\ {A^*}^2 &{} A^* &{} I \end{bmatrix},\) by means of inertia of dilations. PubDate: 2024-06-21 DOI: 10.1007/s43036-024-00360-7
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Abstract: Abstract Isometric covariant representations play an important role in the study of Cuntz–Pimsner algebras. In this article, we study partial isometric covariant representations and explore under what conditions powers and roots of partial isometric covariant representations are also partial isometric covariant representations. PubDate: 2024-06-17 DOI: 10.1007/s43036-024-00359-0
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Abstract: Abstract Let X be a real or complex Banach space of dimension at least 3. We give a complete description of surjective mappings on B(X) that preserve the ascent of Jordan triple product of operators or, preserve the descent of Jordan triple product of operators. PubDate: 2024-06-14 DOI: 10.1007/s43036-024-00357-2
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Abstract: Under an abstract setting, we show that eigenvectors belong to discrete spectra of unitary operators have exponential decay properties. We apply the main theorem to multi-dimensional quantum walks and show that eigenfunctions belong to a discrete spectrum decay exponentially at infinity. PubDate: 2024-06-08 DOI: 10.1007/s43036-024-00358-1
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Abstract: Abstract This work focuses on the investigation of a quasilinear elliptic problem in the entire space \(\mathbb {R}^N\) , which involves the 1-Laplacian and 1-biharmonic operators, as well as potentials that can vanish at infinity. This research is conducted within the space of functions with bounded variation. The main result is proven using a version of the mountain pass theorem that does not require the Palais-Smale condition. PubDate: 2024-06-06 DOI: 10.1007/s43036-024-00351-8
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Abstract: Abstract In this paper, we are starting to construct a new theory of absolutely simple p-summing operators. We define a significant class of weak operator ideals, namely the class of absolutely simple p-summing operators between arbitrary real Banach spaces and show some basic properties of that class. A key feature of the resulting class is computing simple p-summing norms exactly for any linear operator between finite-dimensional normed spaces, in contrast to the computation of p-summing norms which is in general difficulty or the computation of Lipschitz p-summing norms between particular classes of metric spaces. Building upon S. Kwapień’s result, we figure out the relations between 2-summing norms and simple 2-summing norms and find out the relations between simple p-summing norms and diverse familiar norms of some linear operators. In the end, we present some concluding remarks and introduce some open problems that we think are intriguing. PubDate: 2024-06-06 DOI: 10.1007/s43036-024-00356-3
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Abstract: Abstract In this paper the boundary generating curves and numerical ranges of centrosymmetric matrices of orders up to 6 are characterized in terms of the matrices entries. These results extend previous ones concerning Kac-Sylvester matrices. The classification of all the possible boundary generating curves for centrosymmetric matrices of higher dimensions remains open. PubDate: 2024-06-04 DOI: 10.1007/s43036-024-00353-6
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