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Abstract: Abstract Let \( T \in \mathcal {B}(\mathcal {H})\) be a bounded linear operator on a Hilbert space \( \mathcal {H}\) , and let \( T = U \vert T \vert \) be the polar decomposition of T. For any \(r > 0\) , the transform \(S_{r}(T)\) is defined by \(S_{r}(T) = U \vert T \vert ^{r} U\) . In this paper, we discuss the transform \(S_{r}(T)\) of some classes of operators such as p-hyponormal and rank one operators. We provide a new characterization of invertible normal operators via this transform. Afterwards, we investigate when an operator T and its transform \( S_{r}(T) \) both have closed ranges, and show that this transform preserves the class of EP operators. Finally, we present some relationships between an EP operator T, its transform \( S_{r}(T)\) and the Moore–Penrose inverse \( T^{+} \) . PubDate: 2024-02-22

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Abstract: Abstract In this research, we analyze the existence of infinite sequences of ordered solutions for a class of non-local elliptic problem with Dirichlet boundary condition. The primary techniques employed consist of topological degree theory for mappings of type \(S_+\) and minimization arguments in a fractional Orlicz–Sobolev space. Our main results generalize some recent findings in the literature to non-smooth cases. PubDate: 2024-01-23

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Abstract: Abstract The central focus of this paper is the analysis of optimal dual frames for a given frame as well as optimal dual pairs, in light of a probability model-based erasure during the transmission of the frame coefficients corresponding to the data. We consider these two broad and different contexts of the erasure problem and analyze each of them, with the optimality measure taken to be the spectral radius as well as the operator norm of the associated error operators. PubDate: 2024-01-23

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Abstract: Abstract We follow several approaches in nonlinear spectral theory and determine the various spectral forms for the nonlinear weighted superposition operator on Fock spaces. The results show that most of the forms introduced so far coincide and contain singeltons. The classical, asymptotic, and connected eigenvalues, and some numerical ranges of the operator are also identified. We further prove that the operator is both linear and odd asymptotically with respect to the pointwise multiplication operator on the spaces. PubDate: 2024-01-20

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Abstract: Abstract In this paper, using Hardy–Littlewood maximal function, we deal with the boundedness of the \(\psi \) -Riemann–Liouville in Lebesgue and weighted Lebesgue space in the real line. Moreover, we consider the boundedness of \(\psi \) -Riemann–Liouville tempered fractional integrals in weighted Lebesgue space in the real line. PubDate: 2024-01-18

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Abstract: Abstract In this paper, we obtain some new matrix inequalities involving Hadamard product. Also, some Hadamard product inequalities for accretive matrices involving the matrix means, positive unital linear maps, and matrix concave functions are investigated. PubDate: 2024-01-17

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Abstract: Abstract Let S be a n-by-n truncated shift whose numerical radius equal one. First, Cassier et al. (J Oper Theory 80(2):453–480, 2018) proved that the Harnack part of S is trivial if \(n=2\) , while if \(n=3\) , then it is an orbit associated with the action of a group of unitary diagonal matrices; see Theorem 3.1 and Theorem 3.3 in the same paper. Second, Cassier and Benharrat (Linear Multilinear Algebra 70(5):974–992, 2022) described elements of the Harnack part of the truncated n-by-n shift S under an extra assumption. In Sect. 2, we present useful results in the general finite-dimensional situation. In Sect. 3, we give a complete description of the Harnack part of S for \(n=4\) , the answer is surprising and instructive. It shows that even when the dimension is an even number, the Harnack part is bigger than conjectured in Question 2 and we also give a negative answer to Question 1 (the two questions are contained in the last cited paper), when \(\rho =2\) . PubDate: 2023-12-19

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Abstract: Abstract In this article, we generalize an approximation result known for entropy numbers of operators. We show that the entropy numbers of a bounded linear operator can be approximated by those of certain truncations of the operator under very general assumptions. Using the relation between the entropy numbers and the inner entropy numbers of bounded sets, we derive estimates for entropy numbers of bounded linear operators. We also obtain new estimates for specific types of operators, including the diagonal operators between sequence spaces, and use these estimates to illustrate the convergence result proved. PubDate: 2023-12-13

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Abstract: Abstract The Krein transform is the real counterpart of the Cayley transform and gives a one-to-one correspondence between the positive relations and symmetric contractions. It is treated with a slight variation of the usual one, resulting in an involution for linear relations. On the other hand, a semi-bounded linear relation has closed semi-bounded symmetric extensions with semi-bounded selfadjoint extensions. A self-consistent theory of semi-bounded symmetric extensions of semi-bounded linear relations is presented. Using the Krein transform, a formula of positive extensions of quasi-null relations is provided. PubDate: 2023-12-12

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Abstract: Abstract We prove that the closure of the numerical range of a \((n+1)\) -periodic and \((2m+1)\) -banded Toeplitz operator can be expressed as the closure of the convex hull of the uncountable union of numerical ranges of certain symbol matrices. In contrast to the periodic 3-banded (or tridiagonal) case, we show an example of a 2-periodic and 5-banded Toeplitz operator such that the closure of its numerical range is not equal to the numerical range of a single finite matrix. PubDate: 2023-12-11

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Abstract: Abstract Are explored the spectral properties for an unbounded operator U for which there exists an injective quasi-nilpotent unbounded operator N such that \(UN=NU\) . Several important properties in spectral theory, in this new set-up are considered. PubDate: 2023-12-11

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Abstract: Abstract The main goal of this work is to establish further improved inequalities of real power form for Alzer–Fonseca–Kovačec’s inequalities, which extend many key recent results. Some applications to estimate new relative operator entropy inequalities are given. We also propose new refining inequalities for the p-norms, traces, and determinants of \(\tau \) -measurable operators. PubDate: 2023-12-05 DOI: 10.1007/s43036-023-00306-5

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Abstract: Abstract For an n-by-n matrix A partitioned as \(\left[ \begin{array}{ll} B &{} C \\ D &{} E \end{array}\right] \) with B of size m ( \(1\le m<n\) ), we consider various conditions on A and B which guarantee that \(C=0\) and \(D=0\) . Such conditions involve eigenvalues, Kippenhahn polynomials or numerical ranges of A and B. One of them says that if \(\Vert A\Vert =1\) , B is of class \(S_m\) , and \(W(A)=W(B)\) , then we always have \(C=0\) and \(D=0\) . There are others involving A, B, and E: if, for some \(\theta _1\) and \(\theta _2\) in \([0, 2\pi )\) with \( \theta _1-\theta _2 \ne 0, \pi \) , the spectrum of \(\textrm{Re}\,(e^{i\theta _j}A)\) equals the union of the spectra of \(\textrm{Re}\,(e^{i\theta _j}B)\) and \(\textrm{Re}\,(e^{i\theta _j}E)\) for both \(j=1\) and 2, then \(C=0\) and \(D=0\) . PubDate: 2023-11-24 DOI: 10.1007/s43036-023-00303-8

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Abstract: Abstract Supposing the non-positive function V(x) on \({\mathbb {R}}^n\) belongs to \(L_{\text {loc}}^p({\mathbb {R}}^n)\) for some \(p\ge 1,\) we studied the number of negative eigenvalues of higher-order Schrödinger type operators \(L=(-\triangle )^m+V\) with the integer \(2\le m < n/2.\) PubDate: 2023-11-11 DOI: 10.1007/s43036-023-00302-9

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Abstract: Abstract In the paper, it is constructed the universal pair ( \(\mathbb {U}\) , \(\mathbb {E}\) ) in sense of modifications with respect to the generalized Walsh system. PubDate: 2023-10-25 DOI: 10.1007/s43036-023-00301-w

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Abstract: Abstract Suppose that \(\alpha \) is an action of the semigroup \({\mathbb {N}}^{2}\) on a \(C^*\) -algebra A by endomorphisms. Let \(A\times _{\alpha }^{\textrm{piso}} {\mathbb {N}}^{2}\) be the associated partial-isometric crossed product. By applying an earlier result which embeds this semigroup crossed product (as a full corner) in a crossed product by the group \({\mathbb {Z}}^{2}\) , a composition series \(0\le L_{1}\le L_{2}\le A\times _{\alpha }^{\textrm{piso}} {\mathbb {N}}^{2}\) of essential ideals is obtained for which we identify the subquotients with familiar algebras. PubDate: 2023-10-20 DOI: 10.1007/s43036-023-00300-x

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Abstract: Abstract This work studies the inhomogeneous bi-harmonic nonlinear Schrödinger (IBNLS) equation $$\begin{aligned} i u_t+\Delta ^2 u=\pm F(x,u), \quad u:=u(t,x):{\mathbb {R}}\times {\mathbb {R}}^N\rightarrow {\mathbb {C}}. \end{aligned}$$ Here \(N\ge 5\) , \(F(x,u)\in \{ x ^{-2\tau } u ^{2(q-1)}u, x ^{-\tau } u ^{p-2}(J_\alpha * \cdot ^{-\tau } u ^p)u\}\) , where \(p\ge 2\) and \(q>1\) . The Riesz potential is \(J_\alpha (x)=C_{N,\alpha } x ^{-(N-\alpha )}\) , for certain \(0<\alpha <N\) . Moreover, one considers the energy critical regime \(q=1+\frac{4-2\tau }{N-4}\) and \(p=1+\frac{4-2\tau +\alpha }{N-4}\) . The purpose is two-fold. First, one develops a local theory in the energy space. Second, the local solution extends to a global one which scatters for small datum. The novelty here is the investigation of the energy-critical regime. In the local theory, one uses a fix point argument via the fractional Hardy type inequality \(\Vert x ^{-\tau }u\Vert _r\le c\Vert u\Vert _{\dot{W}^{\tau ,r}}\) and Strichartz estimates. Compared with the recent work by An and Kim (A note on the \(H^s\) -critical inhomogeneous nonlinear Schrödinger equation, https://doi.org/10.48550/arXiv.2112.11690), the novelty is twice. First, one considers the IBNLS in the energy-critical regime in classical Sobolev spaces and without using Lorentz spaces. Moreover, one treats the non-local case, namely a Hartree-type source term. Furthermore, in the recent paper by the first author (Local Well-Posedness of a Critical Inhomogeneous Bi-harmonic Schrödinger Equ Mediterr J Math 20:170, 2023), one treats the same problem using a different method based on weighted Lebesgue spaces for a local source term. PubDate: 2023-10-17 DOI: 10.1007/s43036-023-00297-3

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Abstract: Abstract The joint numerical range of n-ple of Hermitian matrices is the linear image of the density matrices. We analyze the boundary of the joint numerical range using the algebraic geometric method based on the dual variety of the irreducible hypersurface of the Kippenhahn polynomial of Hermitian matrices, especially when the dual variety is degenerate. Although the results in this paper deal with specific examples, the computation algorithm is applicable to more general case. PubDate: 2023-10-09 DOI: 10.1007/s43036-023-00299-1

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Abstract: Abstract The purpose of this work is the investigation of a degenerate parabolic equation with double-phase operator, variable growth and strongly nonlinear source under Dirichlet boundary conditions, the existence of a periodic nonnegative weak solution is established. Our proof will be based on the Leray–Schauder topological degree, which presents several issues for this kind of equations, but were overcome using different techniques and known theorems. The system considered is a possible model for problems where the entity studied has different growth exponents, p(x) and q(x) in our case, that varies with the position where the growth is calculated. PubDate: 2023-10-08 DOI: 10.1007/s43036-023-00296-4