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 Advances in Operator TheoryNumber of Followers: 4      Hybrid journal (It can contain Open Access articles) ISSN (Print) 2662-2009 - ISSN (Online) 2538-225X Published by Springer-Verlag  [2658 journals]
• The semicircular law of free probability as noncommutative multivariable
operator theory

Abstract: Abstract In this paper, we study semicircular elements induced by connected finite directed graphs. It is shown that if the graph groupoid $$\mathbb {G}$$ of a given graph G contains at least one loop finite path, then it induces a semicircular element under suitable representations of $$\mathbb {G}$$ . As application, if a graph G is fractal (or, satisfies the fractal property) in a certain sense, then it automatically generates infinitely many semicircular elements.
PubDate: 2021-10-14

• Multiresolution analysis for linear canonical S transform

Abstract: Abstract To deal with the time-varying signals, linear canonical S transform (LCST) is introduced to possess some desirable characteristics that are absent in conventional time–frequency transforms. Inspired by LCST, we in this paper developed an idea of novel MRA associated with LCST. Moreover, the construction method of orthogonal wavelets is developed. Finally an example is provided to justify the results.
PubDate: 2021-10-04

• Lipschitz p-lattice summing operators

Abstract: Abstract In this paper, we introduce and study the notion of Lipschitz p-lattice summing operators in the category of Lipschitz operators which generalizes the class of p-lattice summing operators in the linear case. Some interesting properties are given. Also, some connections with other classes of operators are presented.
PubDate: 2021-09-27

• Analogue of slant Hankel operators on the Lebesgue space of n-torus

Abstract: Abstract In this paper, the multivariate analogue of slant Hankel operator on $$L^2(\mathbb {T}^n)$$ , ( $$n\ge 1$$ , a natural number), the Lebesgue space of square integrable functions defined on $$\mathbb {T}^n$$ , where $$\mathbb {T}$$ is the unit circle, is introduced. Various characterizations are obtained for a bounded operator on $$L^2(\mathbb {T}^n)$$   to be a kth- order slant Hankel operator  ( $$k\ge 2$$ , a fixed integer).
PubDate: 2021-09-07

• Fractional Sobolev space with Riemann–Liouville fractional derivative
and application to a fractional concave–convex problem

Abstract: Abstract A new fractional function space $$E_{L}^{\alpha }[a,b]$$ with Riemann–Liouville fractional derivative and its related properties are established in this paper. Under this configuration, the following fractional concave–convex problem: 0.1 \begin{aligned} \begin{aligned}&{_{x}}D_{b}^{\alpha }({_{a}}D_{x}^{\alpha }u) = \lambda u^\sigma + u^p,\;\;\text{ in }\;\;(a,b)\\&B_{\alpha }(u)=0,\;\;\text{ in }\;\;\partial (a,b) \end{aligned} \end{aligned} where $$\alpha \in (0,1)$$ , $$\sigma \in (0,1)$$ and $$p\in (1, \frac{1+ 2\alpha }{1-2\alpha })$$ if $$\alpha \in (0, \frac{1}{2})$$ and $$p\in (1, +\infty )$$ if $$\alpha \in (\frac{1}{2}, 1)$$ . $$B_\alpha (u)$$ represent the boundary condition of the problem which depend of the behavior of $$\alpha \in (0,1)$$ , that is: \begin{aligned} B_\alpha (u) = {\left\{ \begin{array}{ll} \lim _{x\rightarrow a^+}{_{a}}I_{x}^{1-\alpha }u(x) = 0,&{}\hbox { if}\ \alpha \in \left(0, \frac{1}{2}\right)\\ u(a) = u(b) = 0,&{}\hbox { if}\ \alpha \in \left(\frac{1}{2}, 1\right). \end{array}\right. } \end{aligned} By using Ekeland’s variational principle and mountain pass theorem we show that the problem (0.1) at less has two nontrivial weak solutions.
PubDate: 2021-08-17

• Singular value and unitarily invariant norm inequalities for sums and
products of operators

Abstract: Abstract In this note, we mainly investigate singular value and unitarily invariant norm inequalities for sums and products of operators. First, we present singular value inequality for the quantity $$AX+YB$$ : let A, B, X and $$Y \in B({\mathcal {H}})$$ such that both A and B are positive operators. Then \begin{aligned} s_{j}\left( (AX+YB)\oplus 0\right) \le s_{j}\left( (K+M)\oplus (L_{ 1} + N)\right) , \end{aligned} for $$j = 1,2,\ldots$$ , where $$K=\frac{1}{2}A+\frac{1}{2}A^{\frac{1}{2}} X^{*} ^{2}A^{\frac{1}{2}}$$ , $$L_{1}=\frac{1}{2}B+\frac{1}{2}B^{\frac{1}{2}} Y ^{2}B^{\frac{1}{2}}$$ , $$M=\frac{1}{2} B^{\frac{1}{2}}(X+Y)^{*}A^{\frac{1}{2}}$$ and $$N=\frac{1}{2} A^{\frac{1}{2}}(X+Y)B^{\frac{1}{2}}$$ . In addition, based on the above singular value inequality, we establish a unitarily invariant norm inequality for concave functions. These results generalize inequalities obtained by Audeh directly. Finally, we present another more general singular value inequality for $$\sum \nolimits _{i=1}^{m}A_{i}^{*}X_{i}^{*}B_{i}$$ : \begin{aligned} s_{j}\left( \sum \limits _{i=1}^{m}A_{i}^{*}X_{i}^{*}B_{i}\oplus 0\right) \le s_{j} \left( \left( \sum \limits _{i=1}^{m}A_{i}^{*}f_{i}^{2}( X_{i} )A_{i}\right) \oplus \left( \sum \limits _{i=1}^{m}B_{i}^{*}g_{i}^{2}( X_{i}^{*} )B_{i}\right) \right) , \end{aligned} for $$j=1,2,\ldots$$ , where $$A_{i}$$ , $$B_{i}$$ and $$X_{i}\in B({\mathcal {H}})$$ such that $$A_{i}$$ and $$B_{i}$$ ( $$i=1,2,\ldots ,m$$ ) are compact operators and $$f_{i}$$ , $$g_{i}$$ ( $$i=1,2,\ldots ,m$$ ) are 2m nonnegative continuous functions on $$[0,+\infty )$$ with $$f_{i}(t)g_{i}(t)=t$$ ( $$i=1,2,\ldots ,m$$ ) for $$t\in [0,+\infty )$$ .
PubDate: 2021-08-11

• A caricature of dilation theory

Abstract: Abstract We present a set-theoretic version of some basic dilation results of operator theory. The results we have considered are Wold decomposition, Halmos dilation, Sz. Nagy dilation, inter-twining lifting, commuting and non-commuting dilations, BCL theorem, etc. We point out some natural generalizations and variations.
PubDate: 2021-08-05

• On extension of Beurling–Helson–Lowdenslager theorem

Abstract: Abstract In this paper, we prove an extension of the Helson–Lowdenslager theorem for $$\mathcal {L}^2(\mathbb {T})$$ for sub - Hilbert spaces under certain axioms. Furthermore, we provide an alternate proof of characterization of shift invariant sub-Hilbert spaces contained in $$\mathcal {H}^2(\mathbb {D})$$ satisfying similar axioms.
PubDate: 2021-07-29

• Inertial extragradient algorithms with non-monotonic step sizes for
solving variational inequalities and fixed point problems

Abstract: Abstract In this paper, we introduce four inertial extragradient algorithms with non-monotonic step sizes to find the solution of the convex feasibility problem, which consists of a monotone variational inequality problem and a fixed point problem with a demicontractive mapping. Strong convergence theorems of the suggested algorithms are established under some standard conditions. Finally, we implement some computational tests to show the efficiency and advantages of the proposed algorithms and compare them with some existing ones.
PubDate: 2021-07-19

• Analytic functional calculus for two operators

Abstract: Abstract This paper is a survey devoted to the transformations \begin{aligned} C&\mapsto \frac{1}{(2\pi i)^2}\int _{\Gamma _1}\int _{\Gamma _2}f(\lambda ,\mu )\,R_{1,\,\lambda }\,C\, R_{2,\,\mu }\,{\mathrm{d}}\mu \,{\mathrm{d}}\lambda ,\\ C&\mapsto \frac{1}{2\pi i}\int _{\Gamma }g(\lambda )R_{1,\,\lambda }\,C\, R_{2,\,\lambda }\,{\mathrm{d}}\lambda , \end{aligned} where $$R_{1,\,(\cdot )}$$ and $$R_{2,\,(\cdot )}$$ are pseudo-resolvents acting in a Banach space, i. e., the resolvents of bounded, unbounded, or multivalued linear operators, and f and g are analytic functions; here $$\Gamma _1$$ , $$\Gamma _2$$ , and $$\Gamma$$ surround the singular sets (spectra) of the pseudo-resolvents $$R_{1,\,(\cdot )}$$ , $$R_{2,\,(\cdot )}$$ , and the both, respectively. Several applications are considered: a representation of the impulse response of a second-order linear differential equation with operator coefficients, a representation of the solution of the Sylvester equation, and properties of the differential of the ordinary functional calculus.
PubDate: 2021-07-19

• Rough Hausdorff operator and its commutators on the Heisenberg group

Abstract: Abstract In this paper, we discuss some sufficient conditions for the boundedness of rough Hausdorff operators on the central Morrey and Herz spaces with the Muckenhoupt weights on the Heisenberg group. The boundedness for the commutators of rough Hausdorff operators with symbols in weighted CMO space on such spaces is given as well.
PubDate: 2021-07-07

• Multipliers of double Fourier–Haar series

Abstract: Abstract The multipliers of double Fourier–Haar series for functions from anisotropic Lorentz spaces are investigated. Necessary and sufficient conditions are obtained for the sequence $$\lambda =\{\lambda _{k_1k_2}^{j_1j_2}\}$$ to belong to the class $$m(L_{\bar{p},\bar{r}}\rightarrow L_{\bar{q},\bar{s}})$$ . In particular, the case is described when $$\bar{s}<\bar{r}$$ , which is a new result in the one-dimensional case as well.
PubDate: 2021-07-06

• Weighted laws of large numbers and convergence of weighted ergodic
averages on vector valued $$L_p$$ L p -spaces

Abstract: Abstract A more general notion of weight called admissible is introduced and then an investigation is carried out on the a.e. convergence of weighted strong laws of large numbers and their applications to weighted ergodic averages on vector-valued $$L_p$$ -spaces.
PubDate: 2021-07-05

• Weighted variable Hardy spaces on domains

Abstract: Abstract In this paper, we introduce and study the weighted variable Hardy space on domains. We prove the atomic decompositions of this space, and as application, we figure out its dual space.
PubDate: 2021-06-24

• Explicit formula for MRA-wavelets on local fields

Abstract: Abstract We provide an explicit formula for the wavelets associated with a multiresolution analysis of $$L^2(K)$$ , where K is a local field of positive characteristic. We also give several examples to illustrate this result.
PubDate: 2021-06-17

• Generalised nonlinear Picone identities for p-sub-Laplacians and
p-biharmonic operators and applications

Abstract: Abstract In this paper, we establish generalised nonlinear Picone identities for p-sub-Laplacians, anisotropic p-sub-Laplacians and p-biharmonic operators on general stratified Lie groups. Moreover, we give applications to horizontal Hardy inequalities, Sturmian comparison principles as well as to weighted eigenvalue problems for p-sub-Laplacian and p-biharmonic operators.
PubDate: 2021-06-14

• The Kachurovskij spectrum of Lipschitz continuous nonlinear block operator
matrices

Abstract: Abstract In this paper, the Kachurovskij spectrum of $$2\times 2$$ Lipschitz continuous nonlinear operator matrices are studied. Firstly, some connections between the Kachurovskij spectrum of certain $$2\times 2$$ Lipschitz continuous nonlinear operator matrices and that of their entries are established, and the relationship between the Kachurovskij spectrum of $$2\times 2$$ Lipschitz continuous nonlinear operator matrices and that of their Schur complement is presented by means of Schur decomposition. Then, the Gershgorin’s theorem of $$2\times 2$$ Lipschitz continuous nonlinear operator matrices is given, and the spectral inclusion properties of Lipschitz continuous nonlinear block operator matrices are investigated by using the numerical range of diagonal entries. Finally, the lipeomorphism of certain $$2\times 2$$ Lipschitz continuous nonlinear operator matrices is characterized by using the perturbation theory of nonlinear operators.
PubDate: 2021-06-14

• Completely order bounded maps on non-commutative $${\varvec{L_p}}$$ L p
-spaces

Abstract: Abstract We define norms on $$L_p({\mathcal {M}}) \otimes M_n$$ where $${\mathcal {M}}$$ is a von Neumann algebra and $$M_n$$ is the space of complex $$n \times n$$ matrices. We show that a linear map $$T: L_p({\mathcal {M}}) \rightarrow L_q({\mathcal {N}})$$ is decomposable if $${\mathcal {N}}$$ is an injective von Neumann algebra, the maps $$T \otimes Id_{M_n}$$ have a common upper bound with respect to our defined norms, and $$p = \infty$$ or $$q = 1$$ . For $$2p< q < \infty$$ we give an example of a map $$T$$ with uniformly bounded maps $$T \otimes Id_{M_n}$$ which is not decomposable.
PubDate: 2021-06-08

• Sums of weighted differentiation composition operators from weighted
Bergman spaces to weighted Zygmund and Bloch-type spaces

Abstract: Abstract Let $${\mathcal {H}}({\mathbb {D}})$$ be the space of analytic functions on the unit disc $${\mathbb {D}}$$ and let $${\mathcal {S}}({\mathbb {D}})$$ denote the set of all analytic self maps of the unit disc $${\mathbb {D}}$$ . Let $$\Psi =(\psi _j)_{j=0}^k$$ be such that $$\psi _j\in {\mathcal {H}}({\mathbb {D}})$$ and $$\varphi \in {\mathcal {S}}({\mathbb {D}})$$ . To treat the Stević–Sharma type operators and the products of composition operators, multiplication operators, differentiation operators in a unified manner, Wang et al. considered the following sum operator: \begin{aligned} T_{\Psi ,\varphi }^kf= \sum \limits _{j=0}^k\psi _j\cdot f^{(j)}\circ \varphi = \sum \limits _{j=0}^k{\mathfrak {D}}_{\psi _j,\varphi }^jf, \quad f\in {\mathcal {H}}({\mathbb {D}}). \end{aligned} We characterize the boundedness and compactness of the operators $$T_{\Psi ,\varphi }^k$$ from the weighted Bergman spaces $$A_{v,p}$$ to the weighted Zygmund-type spaces $${\mathcal {Z}}_w$$ and the weighted Bloch-type spaces $${\mathcal {B}}_w$$ . Besides, giving examples of bounded, unbounded, compact and non-compact operators $$T_{\Psi ,\varphi }^k$$ , we give an example of two unbounded weighted differentiation composition operators $${\mathfrak {D}}_{\psi _0,\varphi }^0, \ {\mathfrak {D}}_{\psi _1,\varphi }^1:A_{v,p}\longrightarrow {\mathcal {Z}}_w( {\mathcal {B}}_w)$$ such that their sum operator $${\mathfrak {D}}_{\psi _0,\varphi }^0+ {\mathfrak {D}}_{\psi _1,\varphi }^1= T_{\Psi ,\varphi }^1:A_{v,p}\longrightarrow {\mathcal {Z}}_w( {\mathcal {B}}_w)$$ is bounded.
PubDate: 2021-06-02

• Existence results of renormalized solutions for nonlinear $$p(\cdot )$$ p
( · ) -parabolic equations with possibly singular measure data

Abstract: Abstract We study the existence of renormalized solutions to a nonlinear parabolic boundary value problem with a general and possibly singular measure data, whose model is \begin{aligned} ({\mathcal {P}})\ {\left\{ \begin{array}{ll} \frac{\partial b(u)}{\partial t} -\Delta _{p(x)}u =\mu &{}\ \ \text{ in }\ Q:=\Omega \times (0,T), \\ b(u)(t=0)=b(u_{0})(x) &{}\ \ \text{ in }\ \Omega ,\\ u(x,t)=0 &{}\ \ \text{ on }\ \partial \Omega \times (0,T),\\ \end{array}\right. } \end{aligned} where $$\Omega$$ is an open bounded subset of $${\mathbb {R}}^{N}$$ ( $$N\ge 2$$ ), $$T>0$$ , b is an increasing $$C^{1}$$ -function, $$\Delta _{p(x)}u:=\text {div}( \nabla u ^{p(x)-2}\nabla u)$$ ( $$1<p_{-}\le p(x)\le p_{+}<N$$ ) is the p(x)-Laplacian operator which, roughly speaking, behaves as $$\nabla u ^{p(x)-1}$$ , $$\mu$$ is a bounded Radon measure with bounded total variation on Q and $$b(u_{0})$$ is an integrable function. We provide the assumptions, the notions of solution we are adopting and the statements of the existence result in the “generalized” Sobolev spaces with variable exponent using some “specific” decompositions on the data.
PubDate: 2021-06-01

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