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Abstract: Abstract In this work, we study the combined effect of concave and convex nonlinearities on the number of solutions for a biharmonic elliptic system involving a Rellich-type potential and multiple critical strongly coupled terms. Using the Nehari manifold and analytic techniques, the system is proved to have at least two nontrivial nonnegative solutions. PubDate: 2022-11-27

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Abstract: Abstract The aim of this article is to give a definition of K-frames in Krein spaces. This definition is compatible with K-frames already known in Hilbert spaces and it generalizes them. We will characterize the K-frames by the synthesis operator and the frame operator, likewise to what is seen in the case of Hilbert spaces. In the rest of the article, we will set a definition of dual sequences and some results concerning this notion. Finally, we will demonstrate how to transfer K-frames for Hilbert spaces to Krein spaces arising from a possibly non-regular Gram operator. PubDate: 2022-11-22

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Abstract: Abstract The eigenvalue problems for linear operators emerge in various practical applications in physics and engineering. This paper deals with the eigenvalue problems for the q-Bernstein operators, which play an important role in the q-boson theory of modern theoretical physics. The eigenstructure of the classical Bernstein operators was investigated in detail by S. Cooper and S. Waldron back in 2000. Some of their results were extended for other Bernstein-type operators, including the q-Bernstein and the limit q-Bernstein operators. The current study is a pursuit of this research. The aim of the present work is twofold. First, to derive for the q-Bernstein polynomials analogues of the Cooper–Waldron results on zeroes of the eigenfunctions. Next, to present in detail the proof concerning the existence of non-polynomial eigenfunctions for the limit q-Bernstein operator. PubDate: 2022-11-19

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Abstract: Abstract For a non-negative measure \(\mu\) with p atoms, we study the relation between the Square Root Problem of \(\mu\) and the problem of subnormality of \({{\tilde{W}}_\mu }\) the Aluthge transform of the associated unilateral weighted shift. We use an approach based on uniquely represented elements in the support of \(\mu *\mu\) . We first show that if \({{\tilde{W}}_\mu }\) is subnormal, then \(2p-1\le card(supp(\mu *\mu ))\le [\frac{(p-1)^2+6}{2}]\) . We rewrite several results known for finitely atomic measure having at most five atoms and give a complete solution for measures six atoms. PubDate: 2022-11-18

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Abstract: Abstract In this paper, we give new singular value inequalities for matrices. It is shown that if A, B, X are \(n\times n\) matrices such that X is positive semidefinite, and if \(f:[0,\infty )\rightarrow {\mathbb {R}} \) is an increasing nonnegative convex function, then $$\begin{aligned} s_{j}\left( f\left( \frac{\left AXB^{*}\right }{\left\ X\right\ }\right) \right) \le \frac{\left\ f\left( \frac{A^{*}A+B^{*}B}{2}\right) \right\ }{\left\ X\right\ }s_{j}\left( X\right) \end{aligned}$$ and $$\begin{aligned} s_{j}\left( AXB^{*}\right) \le \frac{1}{2}\left\ \frac{A^{*}A}{ \left\ A\right\ ^{2}}+\frac{B^{*}B}{\left\ B\right\ ^{2}} \right\ \left\ A\right\ \left\ B\right\ s_{j}\left( X\right) \end{aligned}$$ for \(j=1,2,...,n\) . Some of our inequalities present refinements of some known singular value inequalities. PubDate: 2022-11-18

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Abstract: Abstract We study the stability of certain spectra under some algebraic conditions weaker than the commutativity and we generalize many known commutative perturbation results. In particular, in a complex unital Banach algebra \(\mathcal {A},\) we prove that if \(x \in \hbox {comm}_{w}(a)\) and a is nilpotent, then \(\sigma (x)=\sigma (x+a).\) Among other things, we prove that if \(x \in \hbox {comm}(xy)\cup \hbox {comm}(yx)\) or \(y\in \hbox {comm}(yx)\) for all \(x,y \in \mathcal {A},\) then \(\mathcal {A}\) is commutative. When \(\mathcal {A}=L(X)\) is the unital complex Banach algebra of bounded linear operators acting on the complex Banach space X, then \(\sigma _{p}(T){{\setminus }}\{0\}\subset \sigma _{p}(T+N){\setminus }\{0\},\) where N is nilpotent and \(T\in \hbox {comm}(NT).\) PubDate: 2022-11-17

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Abstract: Abstract In this paper, we present certain necessary and sufficient conditions for the existence of a nonzero solution of the operator equation \(XAX=BX.\) We present the general form of the solution and describe all cases when this equation has infinitely many solutions as well as when all the solutions are idempotent. We generalize the existing particular results that concern this equation in the special cases when \(A=B\) or \(A*\le B.\) Furthermore, we derive two possible representations of an arbitrary solution as a \(2\times 2\) operator matrix which allows us to discuss the existence of an invertible and a positive solution, but also to open the discussion to the existence of any type of solutions using related results for \(2\times 2\) operator matrices. Also, we consider the operator equation \(XAX=AX,\) which is closely related to the “invariant subspace problem” and describe the set of all right(left) invertible and all Fredholm solutions. PubDate: 2022-11-09

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Abstract: Abstract The primary objective of this article is to investigate the Monge–Ampère type equation \(\text {det}^{\frac{1}{n}}(D^2u+A(x,u,Du))=f(u)\) in \({\mathbb {R}}^n\) . Among others, the necessary and sufficient condition on f for the existence of entire subsolutions is established. This is probably the first time the necessary and sufficient conditions for the existence of entire subsolutions of Monge–Ampère type equations with A depending on Du has been studied. PubDate: 2022-11-04

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Abstract: Abstract We investigate the following second-order problem $$\begin{aligned} \left\{ \begin{array}{ll} -u''(t) +a(t)u(t)=\lambda b(t)f(u(t)), \ \ \ \ t\in {\mathbb {R}}, \qquad \qquad \qquad \qquad (P) \\ \underset{ t \rightarrow \infty }{\lim }u(t)=0, \end{array} \right. \end{aligned}$$ where \(\lambda >0\) is a parameter, \(a \in C({\mathbb {R}}, (0,\infty )),~b\in C({\mathbb {R}}, [0,\infty ))\) such that \(\underset{ t \rightarrow \infty }{\lim }\frac{b(t)}{a(t)} = 0, ~f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function with \(sf(s)> 0\) for \(s\ne 0\) . For the linear case, i.e., \(f(u)=u\) , we investigate the existence of principal eigenvalue of (P). For the nonlinear case, depending on the behavior of f near 0 and \(\infty\) , we obtain asymptotic behavior and the existence of homoclinic solutions of (P). The proof of our main results is based upon bifurcation technique. PubDate: 2022-11-03

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Abstract: Abstract We study the compact intertwining relations for the multiplication operators between Hardy spaces and Bergman spaces in the unit ball of \({\mathbb{C}}^n.\) PubDate: 2022-10-26

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Abstract: Abstract Let \(L=-\Delta +V\) be a Schrödinger operator with the potential V belonging to the reverse Hölder class \(B_{q}, q>n/2\) . Denote by \(\mathrm{CMO}_{\theta }(\rho )\) the vanishing mean oscillation type space associated with L. By the aid of the regularity estimates of the fractional heat kernel related with L, we investigate the weighted boundedness and compactness of the commutators of operators generated by fractional heat semigroups related to L and functions belonging to \(\mathrm{CMO}_{\theta }(\rho )\) . PubDate: 2022-10-16

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Abstract: Abstract In this paper we introduce the notion of generalized scale of Banach spaces. The spaces of a generalized scale of Banach spaces define a projective limit space denoted by \({\mathcal {X}}_{\Delta }\) . In that space we will define a class of operators based on the degree of nondensifiability. Such operators will be called DND-condensing operators. It is shown that these operators have a fixed point in \({\mathcal {X}}_{\Delta }\) . In this manner we obtain another method for the theory of fixed point that joins the already existing one of measures of noncompactness. Some applications of the DND-condensing operator method will be illustrated. PubDate: 2022-10-15

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Abstract: Abstract The skewness of Banach spaces was introduced by Fitzpatrick–Reznick. In this paper, we compute the skewness \(s(\ell _p-\ell _1)\) of Day–James spaces \(\ell _p-\ell _1\) , where \(1< p < \infty\) . This gives that the inequality \(s(X)\le 2 \rho _X(1)\) is strict for such space X, where \(\rho _X\) is the modulus of smoothness of X. PubDate: 2022-10-13

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Abstract: Abstract Let E be a Banach space with a Schauder basis. We investigate E-factorable operators and their operator ideals. Some characterizations of E-factorable operators are established and we show that some collections of E-factorable operators can be (quasi) Banach operator ideals. Also, we represent the surjective hull and the injective hull of the quasi normed operator ideal of E-factorable operators. PubDate: 2022-10-10

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Abstract: Abstract Denote by H a finite dimensional Hopf \(\hbox {C}^*\) -algebra, K a Hopf \(*\) -subalgebra of H. Starting with the observable algebra \({\mathscr {A}}_{K}\) in non-equilibrium Hopf spin models, in which \({\mathscr {A}}_{K}\) carries a coaction of relative quantum double D(H, K), the field algebra \({\mathscr {F}}_{K}={\mathscr {A}}_{K} \rtimes \widehat{D(H,K)}\) is obtained, where \(\widehat{D(H,K)}\) is the dual of D(H, K). This paper shows that the Haar integral of D(H, K) admits a faithful conditional expectation \(\varGamma\) from \({\mathscr {F}}_{K}\) onto \({\mathscr {A}}_{K}\) . The index of \(\varGamma\) is calculated by virtue of its quasi-basis provided by the matrix units of \(\widehat{D(H,K)}\) . PubDate: 2022-10-04

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Abstract: Abstract We prove the boundedness of parametrized area integral \(\mu ^{\varrho }_S\) and Littlewood–Paley’s \(g^*_\lambda\) -function \(\mu ^{*, \varrho }_\lambda\) on grand variable Herz spaces \(\dot{K}_{q(\cdot )}^{\alpha (\cdot ), p),\theta }({\mathbb {R}}^n)\) , where the two main indices \(\alpha\) and q are variable exponents. Furthermore, the boundedness of their higher order commutators \([b^m, \mu ^{\varrho }_S]\) and \([b^m, \mu ^{*, \varrho }_\lambda ]\) with \(\mathrm {BMO}\) functions are also established on these spaces. Our results are still new even when \(m=1\) and \(\alpha (\cdot )\equiv \alpha\) is constant. PubDate: 2022-10-02

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Abstract: Abstract We prove the existence of a non-trivial hyperinvariant subspace for several sets of polynomially compact operators. The main results of the paper are: (i) a non-trivial norm closed algebra \(\mathcal {A}\subseteq \mathcal {B}(\mathscr {X})\) which consists of polynomially compact quasinilpotent operators has a non-trivial hyperinvariant subspace; (ii) if there exists a non-zero compact operator in the norm closure of the algebra generated by an operator band \(\mathcal {S}\) , then \(\mathcal {S}\) has a non-trivial hyperinvariant subspace. PubDate: 2022-09-29

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Abstract: Abstract We prove that the weighted Bergman projection \(P_\gamma \) is a bounded operator on the weighted Lebesgue space \(L^p(\Omega , r(x)^\lambda \mathrm{{d}}m(x))\) for a certain range of parameters p, \(\gamma \) and \(\lambda \) . Here \(\Omega \) is a bounded domain in \(\mathbb R^n\) with smooth boundary. This result is used to prove boundedness of \(P_\gamma \) acting on weighted mixed norm space \(L^{p,q}_\alpha (\Omega )\) , again assuming certain conditions on the parameters. We describe the dual of harmonic mixed norm space \(B^{p,q}_\alpha (\Omega )\) for a certain range of parameters. PubDate: 2022-09-28

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Abstract: Abstract Recently, we proposed the notion of a transpose symmetric path of weighted \({\mathfrak{M}}\) -means for a symmetric operator mean \({\mathfrak{M}},\) and also we obtained a family of operator means including the weighted Heron, logarithmic and Heinz means. On the other hand, Kubo, Nakamura, Ohno and Wada discussed a path of operator monotone functions named the Barbour path. In this paper, based on our former results and the Barbour path, we introduce new families of weighted operator means. One of them provides the relations among the weighted Heinz, Lehmer and some fundamental means. PubDate: 2022-09-28