Advances in Operator TheoryAdvances in Operator Theory
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Sat, 24 Jun 2017 02:47:31 +0100FeedCreatorAdvances in Operator Theory
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Feed provided by Advances in Operator Theory. Click to visit.Complex interpolation and non-commutative integration
http://www.aot-math.org/article_42356_0.html
We show that under suitable conditions interpolation between a Banach space and its dual yields a Hilbert space at $theta =frac{1}{2}$. By application of this result to the special case of the non-commutative $L^p$-spaces of Leinert [Int. J. Math. textbf{2} (1991), no. 2, 177--182] and Terp [J. Operator Theory textbf{8} (1982), 327--360] we conclude that $L^2$ is a Hilbert space and that $L^p$ is isometrically isomorphic to the dual of $L^q$ without using the isomorphisms of these spaces to $L^p$-spaces of Hilsum [J. Funct. Anal. textbf{40} (1981), 151--169.] and Haagerup [Colloq. Internat. CNRS, 274, CNRS, Paris, 1979].\Haagerup and Pisier [Canad. J. Math. textbf{41} (1989), no. 5, 882--906.], Pisier [Mem. Amer. Math. Soc. textbf{122} (1996), no. 585, viii+103 pp] and Watbled [C. R. Acad. Sci. Paris, t. 321, S'erie I, p. 1437--1440, 1995] gave conditions under which interpolation between a Banach space and its conjugate dual yields a Hilbert space at $frac{1}{2}$. The result mentioned above when put in ``conjugate form'' extends their results.Thu, 26 Jan 2017 20:30:00 +0100On the weak compactness of Weak* Dunford-Pettis operators on Banach lattices
http://www.aot-math.org/article_44450_4671.html
We characterize Banach lattices on which each positive weak* Dunford--Pettis operator is weakly (resp., M-weakly, resp., order weakly) compact. More precisely, we prove that if $F$ is a Banach lattice with order continuous norm, then each positive weak* Dunford--Pettis operator $T : Elongrightarrow F$ is weakly compact if, and only if, the norm of $E^{prime}$ is order continuous or $F$ is reflexive. On the other hand, when the Banach lattice $F$ is Dedekind $sigma$-complete, we show that every positive weak* Dunford--Pettis operator $T: Elongrightarrow F$ is M-weakly compact if, and only if, the norms of $E^{prime}$ and $F$ are order continuous or $E$ is finite-dimensional.Fri, 30 Jun 2017 19:30:00 +0100Two-weight norm inequalities for the higher-order commutators of fractional integral operators
http://www.aot-math.org/article_44490_4671.html
In this paper, we obtain several sufficient conditions such that the higher-order commutators $I_{alpha,b}^m$ generated by $I_alpha$ and $bin textrm{BMO}(mathbb{R}^n)$ is bounded from $L^p(v)$ to $L^q(u)$, where $frac{1}{q}=frac{1}{p}-frac{alpha}{n}$ and $0<alpha<n$.Fri, 30 Jun 2017 19:30:00 +0100Properties of $J$-fusion frames in Krein spaces
http://www.aot-math.org/article_44491_4671.html
In this article we introduce the notion of $J$-Parseval fusion frames in a Krein space $mathbb{K}$ and characterize 1-uniform $J$-Parseval fusion frames with $zeta=sqrt{2}$. We provide some results regarding construction of new $J$-tight fusion frame from given $J$-tight fusion frames. We also characterize an uniformly $J$-definite subspace of a Krein space $mathbb{K}$ in terms of $J$-fusion frame. Finally we generalize the fundamental identity of Hilbert space frames in the setting of Krein spaces.Fri, 30 Jun 2017 19:30:00 +0100On the behavior at infinity of certain integral operator with positive kernel
http://www.aot-math.org/article_44569_4671.html
Let $alpha>0$ and $gamma>0$. We consider integral operator of the form$${mathcal{G}}_{phi_gamma}f(x):=frac{1}{Psi_gamma (x)}int_0^x (1-frac{y}{x})^{alpha-1}phi_gamma(y) f(y)dy,,,,, x>0.$$This paper is devoted to the study of the infinity behavior of ${mathcal{G}}_{phi_gamma}$. We also provide separately result on the similar problem in the weighted Lebesgue space.Fri, 30 Jun 2017 19:30:00 +0100Non-commutative rational functions in strong convergent random variables
http://www.aot-math.org/article_46452_0.html
Random matrices like GUE, GOE and GSE have been shown that they possess a lot of nice properties. In 2005, a new property of independent GUE random matrices is discovered by Haagerup and ThorbjÃ¸rnsen in their paper in 2005, it is called strong convergence property and then more random matrices with this property are followed. In general, the definition can be stated for a sequence of tuples over some $text{C}^{ast}$-algebras. In this paper, we want to show that, for a sequence of strongly convergent random variables, non-commutative polynomials can be extended to non-commutative rational functions under certain assumptions. As a direct corollary, we can conclude that for a tuple $(X_{1}^{left(nright)},cdots,X_{m}^{left(nright)})$ of independent GUE random matrices, $r(X_{1}^{left(nright)},cdots,X_{m}^{left(nright)})$ converges in trace and in norm to $r(s_{1},cdots,s_{m})$ almost surely, where $r$ is a rational function and $(s_{1},cdots,s_{m})$ is a tuple of freely independent semi-circular elements which lies in the domain of $r$.Sun, 28 May 2017 19:30:00 +0100Equivalent conditions of a Hardy-type integral inequality related to the extended Riemann zeta ...
http://www.aot-math.org/article_44577_4671.html
By the use of techniques of real analysis and weight functions, we obtain two lemmas and build a few equivalent conditions of a Hardy-type integral inequality with a non-homogeneous kernel, related to a parameter where the constant factor is expressed in terms of the extended Riemann zeta function. Meanwhile, a few equivalent conditions for two kinds of Hardy-type integral inequalities with the homogeneous kernel are deduced. We also consider the operator expressions.Fri, 30 Jun 2017 19:30:00 +0100Existence theorems for attractive points of semigroups of Bregman generalized nonspreading ...
http://www.aot-math.org/article_44913_4671.html
In this paper, we establish new attractive point theorems for semigroups of generalized Bregman nonspreading mappings in reflexive Banach spaces. Our theorems improve and extend many results announced recently in the literature.Fri, 30 Jun 2017 19:30:00 +01002-Local derivations on matrix algebras and algebras of measurable operators
http://www.aot-math.org/article_43482_0.html
Let $mathcal{A}$ be a unital Banach algebra such that any Jordan derivation from $mathcal{A}$ into any $mathcal{A}$-bimodule $mathcal{M}$ is a derivation. We prove that any 2-local derivation from the algebra $M_n(mathcal{A})$ into $M_n(mathcal{M}),,(ngeq 3)$ is a derivation. We apply this result to show that any 2-local derivation on the algebra of locally measurable operators affiliated with a von Neumann algebra without direct abelian summands is a derivation.Thu, 23 Feb 2017 20:30:00 +0100Boundedness of multilinear integral operators and their commutators on generalized Morrey spaces
http://www.aot-math.org/article_45124_4671.html
In this paper, we obtain some boundedness of multilinear Calder'on-Zygmund Operators, multilinear fractional integral operators and their commutators on generalized Morrey Spaces.Fri, 30 Jun 2017 19:30:00 +0100Homomorphic conditional expectations as noncommutative retractions
http://www.aot-math.org/article_46633_0.html
Let $A$ be a $C^*$-algebra and $mathcal{E}colon A to A$ a conditional expectation. The Kadison-Schwarz inequality for completely positive maps, $$mathcal{E}(x)^* mathcal{E}(x) leq mathcal{E}(x^* x),$$implies that$$leftVertmathcal{E}(x)rightVert^2 leq leftVertmathcal{E}(x^* x)rightVert.$$In this note we show that $mathcal{E}$ is homomorphic (in the sense that $mathcal{E}(xy) = mathcal{E}(x)mathcal{E}(y)$ for every $x, y$ in $A$) if and only if$$leftVertmathcal{E}(x)rightVert^2 = leftVertmathcal{E}(x^*x)rightVert,$$for every $x$ in $A$. We also prove that a homomorphic conditional expectation on a commutative $C^*$-algebra $C_0(X)$ is given by composition with a continuous retraction of $X$. One may therefore consider homomorphic conditional expectations as noncommutative retractions.Mon, 05 Jun 2017 19:30:00 +0100Semigroup homomorphisms on matrix algebras
http://www.aot-math.org/article_45172_4671.html
We explore the connection between ring homomorphisms and semigroup homomorphisms on matrix algebras over rings or $C^*$-algebras. Further, we give a connection between group homomorphisms on the general linear groups of a matrix stable $C^*$-algebra and their potentially extended homomorphisms on the whole $C^*$-algebra.Fri, 30 Jun 2017 19:30:00 +0100Variants of Weyl's theorem for direct sums of closed linear operators
http://www.aot-math.org/article_46634_0.html
If $T$ is an operator with compact resolvent and $S$ is any densely defined closed linear operator, then the orthogonal direct sum of $T$ and $S$ satisfies various Weyl type theorems if some necessary conditions are imposed on the operator $S$. It is shown that if $S$ is isoloid and satisfies Weyl's theorem, then $T oplus S$ satisfies Weyl's theorem. Analogous result is proved for a-Weyl's theorem. Further, it is shown that Browder's theorem is directly transmitted from $S$ to $T oplus S$. The converse of these results have also been studied.Tue, 06 Jun 2017 19:30:00 +0100Semicontinuity and closed faces of C*-algebras
http://www.aot-math.org/article_43918_0.html
C. Akemann and G.K. Pedersen [Duke Math. J. 40 (1973), 785--795.] defined three concepts of semicontinuity for self-adjoint elements of $A^{**}$, the enveloping von Neumann algebra of a $C^*$-algebra $A$. We give the basic properties of the analogous concepts for elements of $pA^{**}p$, where $p$ is a closed projection in $A^{**}$. In other words, in place of affine functionals on $Q$, the quasi--state space of $A$, we consider functionals on $F(p)$, the closed face of $Q$ suppported by $p$. We prove an interpolation theorem: If $hgeq k$, where $h$ is lower semicontinuous on $F(p)$ and $k$ upper semicontinuous, then there is a continuous affine functional $x$ on $F(p)$ such that $kleq xleq h$. We also prove an interpolation--extension theorem: Now $h$ and $k$ are given on $Q$, $x$ is given on $F(p)$ between $h_{|F(p)}$ and $k_{|F(p)}$, and we seek to extend $x$ to $widetilde x$ on $Q$ so that $kleqwidetilde xleq h$. We give a characterization of $pM(A)_{{text{sa}}}p$ in terms of semicontinuity. And we give new characterizations of operator convexity and strong operator convexity in terms of semicontinuity.Fri, 03 Mar 2017 20:30:00 +0100Applications of ternary rings to $C^*$-algebras
http://www.aot-math.org/article_45350_4671.html
We show that there is a functor from the category of positive admissible ternary rings to the category of $*$-algebras, which induces an isomorphism of partially ordered sets between the families of $C^*$-norms on the ternary ring and its corresponding $*$-algebra. We apply this functor to obtain Morita-Rieffel equivalence results between cross-sectional $C^*$-algebras of Fell bundles, and to extend the theory of tensor products of $C^*$-algebras to the larger category of full Hilbert $C^*$-modules. We prove that, like in the case of $C^*$-algebras, there exist maximal and minimal tensor products. As applications we give simple proofs of the invariance of nuclearity and exactness under Morita-Rieffel equivalence of $C^*$-algebras.Fri, 30 Jun 2017 19:30:00 +0100On orthogonal decomposition of a Sobolev space
http://www.aot-math.org/article_46656_0.html
The theme of this short article is to investigate an orthogonal decomposition of the Sobolev space $W^{1,2}left( Omega right) $ as $ W^{1,2}left( Omega right) =A^{2,2}left( Omega right) oplus D^{2}left( W_{0}^{3,2}left( Omega right) right)$ and look at some of the properties of the inner product therein and the distance defined from the inner product. We also determine the dimension of the orthogonal difference space $W^{1,2}left( Omega right) ominus left(W_{0}^{1,2}left( Omega right) right) ^{perp }$ and show the expansion of Sobolev spaces as their regularity increases.Tue, 06 Jun 2017 19:30:00 +0100The closure of ideals of $\ell^1(\Sigma)$ in its enveloping $\mathrm{C}^*$-algebra
http://www.aot-math.org/article_44047_0.html
If $X$ is a compact Hausdorff space and $sigma$ is a homeomorphism of $X$, then an involutive Banach algebra $ell^1(Sigma)$ of crossed product type is naturally associated with the topological dynamical system $Sigma=(X,sigma)$. We initiate the study of the relation between two-sided ideals of $ell^1(Sigma)$ and ${mathrm C}^ast(Sigma)$, the enveloping $mathrm{C}^ast$-algebra ${mathrm C}(X)rtimes_sigmamathbb Z$ of $ell^1(Sigma)$. Among others, we prove that the closure of a proper two-sided ideal of $ell^1(Sigma)$ in ${mathrm C}^ast(Sigma)$ is again a proper two-sided ideal of ${mathrm C}^ast(Sigma)$.Tue, 07 Mar 2017 20:30:00 +0100$k$th-order slant Toeplitz operators on the Fock space
http://www.aot-math.org/article_46068_4671.html
The notion of slant Toeplitz operators $B_phi$ and $k$th-order slant Toeplitz operators $B_phi^k$ on the Fock space is introduced and some of its properties are investigated. The Berezin transform of slant Toeplitz operator $B_phi$ is also obtained. In addition, the commutativity of $k$th-order slant Toeplitz operators with co-analytic and harmonic symbols is discussed.Fri, 30 Jun 2017 19:30:00 +0100On symmetry of Birkhoff-James orthogonality of linear operators
http://www.aot-math.org/article_46810_0.html
A bounded linear operator $T$ on a normed linear space $mathbb{X}$ is said to be right symmetric (left symmetric) if $Aperp_{B} T Rightarrow T perp_B A $ ($T perp_{B} A Rightarrow A perp_B T $) for all $ A in B(mathbb{X}),$ the space of all bounded linear operators on $mathbb{X}$. Turnsek [Linear Algebra Appl., 407 (2005), 189-195] proved that if $mathbb{X}$ is a Hilbert space then $T$ is right symmetric if and only if $T$ is a scalar multiple of an isometry or coisometry. This result fails in general if the Hilbert space is replaced by a Banach space. The characterization of right and left symmetric operators on a Banach space is still open. In this paper we study the orthogonality in the sense of Birkhoff-James of bounded linear operators on $ (mathbb{R}^n, |cdot|_{infty}) $ and characterize the right symmetric and left symmetric operators on $(mathbb{R}^n,|cdot|_{infty}).$Sun, 11 Jun 2017 19:30:00 +0100Comparison results for proper multisplittings of rectangular matrices
http://www.aot-math.org/article_46077_4671.html
The least square solution of minimum norm of a rectangular linear system of equations can be found out iteratively by using matrix splittings. However, the convergence of such an iteration scheme arising out of a matrix splitting is practically very slow in many cases. Thus, works on improving the speed of the iteration scheme have attracted great interest. In this direction, comparison of the rate of convergence of the iteration schemes produced by two matrix splittings is very useful. But, in the case of matrices having many matrix splittings, this process is time-consuming. The main goal of the current article is to provide a solution to the above issue by using proper multisplittings. To this end, we propose a few comparison theorems for proper weak regular splittings and proper nonnegative splittings first. We then derive convergence and comparison theorems for proper multisplittings with the help of the theory of proper weak regular splittings.Fri, 30 Jun 2017 19:30:00 +0100Fourier Multiplier Norms of Spherical Functions on the Generalized Lorentz Groups
http://www.aot-math.org/article_47035_0.html
Our main result provides a closed expression for the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups $SO_0(1,n)$ (for $ngeq2$). As a corollary, we find that there is no uniform bound on the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups. We extend the latter result to the groups $SU(1,n)$, $Sp(1,n)$ (for $ngeq2$) and the exceptional group $F_{4(-20)}$, and as an application we obtain that each of the above mentioned groups has a completely bounded Fourier multiplier, which is not the coefficient of a uniformly bounded representation of the group on a Hilbert space.Sun, 18 Jun 2017 19:30:00 +0100Positive map as difference of two completely positive or super-positive maps
http://www.aot-math.org/article_44116_0.html
For a linear map from ${mathbb M}_m$ to ${mathbb M}_n$, besides the usual positivity, there are two stronger notions, complete positivity and super positivity. Given a positive linear map $varphi$ we study a decomposition $varphi = varphi^{(1)} - varphi^{(2)}$ with completely positive linear maps $varphi^{(j)} (j = 1,2)$. Here $varphi^{(1)} + varphi^{(2)}$ is of simple form with norm small as possible. The same problem is discussed with super-positivity in place of complete positivity.Fri, 10 Mar 2017 20:30:00 +0100Almost periodicity of abstract Volterra integro-differential equations
http://www.aot-math.org/article_46543_4671.html
The main purpose of this paper is to investigate almost periodic properties of various classes of $(a,k)$-regularized $C$-resolvent families in Banach spaces. We contemplate the work of many other authors working in this field, giving also some original contributions and applications. In general case, $(a,k)$-regularized $C$-resolvent families under our considerations are degenerate and their subgenerators are multivalued linear operators or pairs of closed linear operators. We also consider the class of $(a,k)$-regularized $(C_{1},C_{2})$-existence and uniqueness families, where the operators $C_{1}$ and $C_{2}$ are not necessarily injective, and provide several illustrative examples of abstract Volterra integro-differential equations which do have almost periodic solutions.Fri, 30 Jun 2017 19:30:00 +0100Traces for fractional Sobolev spaces with variable exponents
http://www.aot-math.org/article_47208_0.html
In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if $pcolonoverline{Omega }times overline{Omega } rightarrow (1,infty )$ and $qcolonpartial Omegarightarrow (1,infty )$ are continuous functions such that[ frac{(n-1)p(x,x)}{n-sp(x,x)}>q(x) qquad mbox{ in } partial Omega cap {xinoverline{Omega}colon n-sp(x,x) >0}, ]then the inequality $$ Vert fVert _{scriptstyle L^{q(cdot)}(partial Omega )} leq C left{ Vert fVert _{scriptstyle L^{bar{p}(cdot)}(Omega )}+ [f]_{s,p(cdot,cdot)} right} $$
holds. Here $bar{p}(x)=p(x,x)$ and $lbrack frbrack_{s,p(cdot,cdot)} $ denotes the fractional seminorm with variable exponent, that is given by begin{equation*} lbrack frbrack_{s,p(cdot,cdot)} := inf left{ lambda >0colon int_{Omega}int_{Omega }frac{|f(x)-f(y)|^{p(x,y)}}{lambda ^{p(x,y)} |x-y|^{n+sp(x,y)}}dxdy<1right} end{equation*}and $Vert fVert _{scriptstyle L^{q(cdot)}(partial Omega )}$ and $Vert fVert _{scriptstyle L^{bar{p}(cdot)}(Omega )}$ are the usual Lebesgue norms with variable exponent.Tue, 20 Jun 2017 19:30:00 +0100A note on O-frames for operators
http://www.aot-math.org/article_46574_4671.html
A sufficient condition for a boundedly complete O-frame and a necessary condition for an unconditional O-frame are given. Also, a necessary and sufficient condition for an absolute O-frame is obtained. Finally, it is proved that if two operators have an absolute O-frame, then their product also has an absolute O-frame.Fri, 30 Jun 2017 19:30:00 +0100Some natural subspaces and quotient spaces of $L^1$
http://www.aot-math.org/article_44924_0.html
We show that the space $text{Lip}_0(mathbb R^n)$ is the dual space of $L^{1}({mathbb R}^{n}; {mathbb R}^{n})/N$ where $N$ is the subspace of $L^{1}({mathbb R}^{n}; {mathbb R}^{n})$ consisting of vector fields whose divergence vanishes identically. We prove that although the quotient space $L^{1}({mathbb R}^{n}; {mathbb R}^{n})/N$ is weakly sequentially complete, the subspace $N$ is not nicely placed - in other words, its unit ball is not closed for the topology $tau_m$ of local convergence in measure. We prove that if $Omega$ is a bounded open star-shaped subset of $mathbb {R}^n$ and $X$ is a dilation-stable closed subspace of $L^1(Omega)$ consisting of continuous functions, then the unit ball of $X$ is compact for the compact-open topology on $Omega$. It follows in particular that such spaces $X$, when they have Grothendieck's approximation property, have unconditional finite-dimensional decompositions and are isomorphic to weak*-closed subspaces of $l^1$. Numerous examples are provided where such results apply.Thu, 13 Apr 2017 19:30:00 +0100Partial isometries: a survey
http://www.aot-math.org/article_45165_0.html
We survey the main results characterizing partial isometries in C$^*$-algebras and tripotents in JB$^*$-triples obtained in terms of regularity, conorm, quadratic-conorm, and the geometric structure of the underlying Banach spaces.Mon, 24 Apr 2017 19:30:00 +0100Operators with compatible ranges in an algebra generated by two orthogonal projections
http://www.aot-math.org/article_45166_0.html
The criterion is obtained for operators A from the algebra generated by two orthogonal projections P,Q to have a compatible range, i.e., coincide with the hermitian conjugate of A on the orthogonal complement to the sum of their kernels. In the particular case of A being a polynomial in P,Q, some easily verifiable conditions are derived.Mon, 24 Apr 2017 19:30:00 +0100Permanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin property
http://www.aot-math.org/article_45177_0.html
Let $P subset A$ be an inclusion of unital $C^*$-algebras and $Ecolon A rightarrow P$ be a faithful conditional expectation of index finite type. Suppose that $E$ has the Rokhlin property. Then $dr(P) leq dr(A)$ and $dim_{nuc}(P) leq dim_{nuc}(A)$. This can be applied to Rokhlin actions of finite groups. We also show that under the same above assumption if $A$ is exact and pure, that is, the Cuntz semigroups $W(A)$ has strict comparison and is almost divisible, then $P$ and the basic contruction $C^*langle A, e_Prangle$ are also pure.Wed, 26 Apr 2017 19:30:00 +0100Almost Hadamard matrices with complex entries
http://www.aot-math.org/article_45905_0.html
We discuss an extension of the almost Hadamard matrix formalism, to the case of complex matrices. Quite surprisingly, the situation here is very different from the one in the real case, and our conjectural conclusion is that there should be no such matrices, besides the usual Hadamard ones. We verify this conjecture in a number of situations, and notably for most of the known examples of real almost Hadamard matrices, and for some of their complex extensions. We discuss as well some potential applications of our conjecture, to the general study of complex Hadamard matrices.Thu, 11 May 2017 19:30:00 +0100