Banach Journal of Mathematical Analysis Articles (Project Euclid)

Note on extreme points in Marcinkiewicz function spaces

Thu, 05 Aug 2010 15:41 EDT

Anna Kaminska , Anca M. Parrish

Source: Banach J. Math. Anal., Volume 4, Number 1, 1--12.

Abstract:
We show that the unit ball of the subspace $M_W^0$ of ordered continuous elements of $M_W$ has no extreme points, where $M_W$ is the Marcinkiewicz function space generated by a decreasing weight function $w$ over the interval $(0,\infty)$ and $W(t) = \int_0^tw$, $t\in(0,\infty)$. We also present here a proof of the fact that a function $f$ in the unit ball of $M_W$ is an extreme point if and only if $f^*=w$.

Index computation for amalgamated products of product systems

Sun, 14 Aug 2011 19:03 EDT

Mithun Mukherjee

Source: Banach J. Math. Anal., Volume 5, Number 1, 148--166.

Abstract:
The notion of amalgamation of product systems has been introduced in \cite{BhM} which generalizes the concept of Skeide product, introduced by Skeide, of two product systems via a pair of normalized units. In this paper we show that amalgamation leads to a setup where a product system is generated by two subsystems and conversely whenever a product system is generated by two subsystems, it could be realized as an amalgamated product. We parameterize all contractive morphism from a Type I product system to another Type I product system and compute index of amalgamated product through contractive morphisms.

The geometry of L^p-spaces over atomless measure spaces and the Daugavet property

Sun, 14 Aug 2011 19:03 EDT

Enrique A. Sanchez Perez, Dirk Werner

Source: Banach J. Math. Anal., Volume 5, Number 1, 167--180.

Abstract:
We show that $L^p$-spaces over atomless measure spaces can be characterized in terms of a $p$-concavity type geometric property that is related with the Daugavet property.

Elementary operators and subhomogeneous C*-algebras II

Sun, 14 Aug 2011 19:03 EDT

Ilja Gogic

Source: Banach J. Math. Anal., Volume 5, Number 1, 181--192.

Abstract:
Let $A$ be a separable unital $C^*$-algebra and let $\Theta_A$ be the canonical contraction from the Haagerup tensor product of $A$ with itself to the space of completely bounded maps on $A$. In our previous paper we showed that if $A$ satisfies (a) the lengths of elementary operators on $A$ are uniformly bounded, or (b) the image of $\Theta_A$ equals the set of all elementary operators on $A$, then $A$ is necessarily SFT (subhomogeneous of finite type). In this paper we extend this result; we show that if $A$ satisfies (a) or (b) then the codimensions of $2$-primal ideals of $A$ are also finite and uniformly bounded. Using this, we provide an example of a unital separable SFT algebra which does not satisfy (a) nor (b). However, if the primitive spectrum of a unital SFT algebra $A$ is Hausdorff, we show that such an $A$ satisfies (a) and (b).

Linear maps respecting unitary conjugation

Sun, 14 Aug 2011 19:03 EDT

B. V. Rajarama Bhat

Source: Banach J. Math. Anal., Volume 5, Number 2, 1--5.

Abstract:
We characterize linear maps on von Neumann algebras which leave every unital subalgebra invariant. We use this characterization to determine linear maps which respect unitary conjugation, answering a question of M. S. Moslehian.

Quasi-multipliers of the dual of a Banach algebra

Sun, 14 Aug 2011 19:03 EDT

M. Adib, A. Riazi, J. Bracic

Source: Banach J. Math. Anal., Volume 5, Number 2, 6--14.

Abstract:
In this paper we extend the notion of quasi-multipliers to the dual of a Banach algebra $A$ whose second dual has a mixed identity. We consider algebras satisfying weaker condition than Arens regularity. Among others we prove that for an Arens regular Banach algebra which has a bounded approximate identity the space $QM_{r}(A^{*})$ of all bilinear and separately continuous right quasi-multipliers of $A^{*}$ is isometrically isomorphic to $A^{**}.$ We discuss the strict topology on $QM_{r}(A^{*})$ and apply our results to $C^{*}-$algebras and to the group algebra of a compact group.

A Fixed point theorem on cone metric spaces with new type contractivity

Sun, 14 Aug 2011 19:03 EDT

Ishak Altun, Mujahid Abbas, Hakan Simsek

Source: Banach J. Math. Anal., Volume 5, Number 2, 15--24.

Abstract:
In the present work, a common fixed point theorem for self maps on cone metric spaces is proved. Also two examples, which shows that our main theorem is generalized version of main theorems of [A. Branciari, Int. J. Math. Math. Sci., 29 (2002), no. 9, 531-536] and [L.G. Huang and X. Zhang, J. Math. Anal. Appl. 332 (2007), no. 2, 1468-1476] are given.

EP elements in Banach algebras

Sun, 14 Aug 2011 19:03 EDT

Dijana Mosic, Dragan S. Djordjevic

Source: Banach J. Math. Anal., Volume 5, Number 2, 25--32.

Abstract:
An element of a Banach algebra is EP, if it commutes with its Moore-Penrose inverse. We present a number of new characterizations of EP elements in Banach algebra.

Multiple Hilbert's type inequalities with a homogeneous kernel

Sun, 14 Aug 2011 19:03 EDT

Ivan Peric, Predrag Vukovic

Source: Banach J. Math. Anal., Volume 5, Number 2, 33--43.

Abstract:
The main objective of this paper is a study of some new generalizations of Hilbert's and Hardy-Hilbert's type inequalities. We apply our general results to homogeneous functions. Also, we obtain the best possible constants when the parameters satisfy appropriate conditions.

Primitivity of some full group C*-algebras

Sun, 14 Aug 2011 19:03 EDT

Erik Bedos, Tron A. Omland

Source: Banach J. Math. Anal., Volume 5, Number 2, 44--58.

Abstract:
We show that the full group $C^*$-algebra of the free product of two nontrivial countable amenable discrete groups, where at least one of them has more than two elements, is primitive. We also show that in many cases, this $C^*$-algebra is antiliminary and has an uncountable family of pairwise inequivalent, faithful irreducible representations.

On pseudodifferential operators with symbols in generalized Shubin classes and an application to Landau-Weyl operators

Sun, 14 Aug 2011 19:03 EDT

Franz Luef, Zohreh Rahbani

Source: Banach J. Math. Anal., Volume 5, Number 2, 59--72.

Abstract:
The relevance of modulation spaces for deformation quantization, Landau--Weyl quantization and noncommutative quantum mechanics became clear in recent work. We continue this line of research and demonstrate that $Q_s(\mathbb{R}^{2d})$ is a good class of symbols for Landau-Weyl quantization and propose that the modulation spaces $M^p_{v_s}(\mathbb{R}^{2d})$ are natural generalized Shubin classes for the Weyl calculus. This is motivated by the fact that the Shubin class $Q_s(\mathbb{R}^{2d})$ is the modulation space $M^2_{v_s}(\mathbb{R}^{2d})$. The main result gives estimates of the singular values of pseudodifferential operators with symbols in $M^p_{v_s}(\mathbb{R}^{2d})$ for the standard Weyl calculus and for the Landau--Weyl calculus.

Stabilizing isomorphisms from $\ell_{p}(\ell_{2})$ into $L_p[0,1]$

Sun, 14 Aug 2011 19:03 EDT

Ran Levy, Gideon Schechtman

Source: Banach J. Math. Anal., Volume 5, Number 2, 73--83.

Abstract:
Let $1< p\neq 2<\infty$, $\e>0$ and let $T$ be an isomorphism from $\ell_p(\ell_2)$ into $L_p[0,1]$. Then there is a subspace $Y\subset \ell_p(\ell_2)$, $(1+\e)$-isomorphic to $\ell_p(\ell_2)$ such that $T_{|Y}$ is an $(1+\e)$-isomorphism and $T\left(Y\right)$ is $K_p$-complemented in $L_{p}\left[0,1\right]$, with $K_p$ depending only on $p$. Moreover, $K_p\le (1+\e)\gamma_p$ if $p>2$ and $K_p\le (1+\e)\gamma_{p/(p-1)}$ if $1<p<2$, where $\gamma_r$ is the $L_r$ norm of a standard Gaussian variable.

The Gelfand--Phillips property in closed subspaces of some operator spaces

Sun, 14 Aug 2011 19:03 EDT

Manijeh Salimi, S. Mohammad Moshtaghioun

Source: Banach J. Math. Anal., Volume 5, Number 2, 84--92.

Abstract:
By introducing the concept of limited completely continuous operators between two arbitrary Banach spaces $X$ and $Y$, we give some properties of this concept related to some well known classes of operators and specially, related to the Gelfand-Phillips property of the space $X$ or $Y$. Then some necessary and sufficient conditions for the Gelfand--Phillips property of closed subspace $M$ of some operator spaces, with respect to limited complete continuity of some operators on $M$, so-called, evaluation operators, are verified.

On strict inclusion relations between approximation and interpolation spaces

Sun, 14 Aug 2011 19:03 EDT

Jose Maria Almira

Source: Banach J. Math. Anal., Volume 5, Number 2, 93--105.

Abstract:
Approximation spaces, in their many presentations, are well known mathematical objects and many authors have studied them for long time. They were introduced by Butzer and Scherer in 1968 and, independently, by Y. Brudnyi and N. Kruglyak in 1978, and popularized by Pietsch in his seminal paper of 1981. Pietsch was interested in the parallelism that exists between the theories of approximation spaces and interpolation spaces, so that he proved embedding, reiteration and representation results for approximation spaces. In particular, embedding results are a natural part of the theory since its inception. The main goal of this paper is to prove that, for certain classes of approximation schemes $(X,\{A_n\})$ and sequence spaces $S$, if $S_1\subset S_2\subset c_0$ (with strict inclusions) then the approximation space $A(X,S_1,\{A_n\})$ is properly contained into $A(X,S_2,\{A_n\})$. We also initiate a study of strict inclusions between interpolation spaces, for Petree's real interpolation method.

Functional decomposition of state induced C*-matrix spaces

Sun, 14 Aug 2011 19:03 EDT

Titarii Wootijirattikal, Sing-Cheong Ong

Source: Banach J. Math. Anal., Volume 5, Number 2, 106--121.

Abstract:
A theorem of Dixmier states that each bounded linear functional $f$ on the algebra of bounded linear operators on a separable Hilbert space is a direct sum of a trace functional $g$ and a singular functional $h$, vanishing on the compact operators, such that $\N f= \N g+\N h$. We use elementary methods to construct, via the state space of a $C\sp*$-algebra, a Banach space of $C\sp*$ matrices that contains a closed subspace on which a version of Dixmier's theorem is proved. When the $C\sp*$-algebra is taken to be the complex numbers our approach gives elementary and transparent proofs of Dixmier's theorem and the trace formula $\rm{tr}(AB) = \rm{tr}(BA)$, without using the operator theoretical machineries used in the known proofs.

C*-reflexivity doesn't pass to quotients

Sun, 14 Aug 2011 19:03 EDT

Yannis Tsertos

Source: Banach J. Math. Anal., Volume 5, Number 2, 122--125.

Abstract:
Using a recently obtained criterion of $C^*$-reflexivity for commutative $C^*$-algebras, we show that the $C^*$-algebra of continuous functions on the Higson corona is not $C^*$-reflexive. This implies that $C^*$-reflexivity doesn't pass to quotient $C^*$-algebras.

Cosine functions revisited

Sun, 14 Aug 2011 19:03 EDT

Dilian Yang

Source: Banach J. Math. Anal., Volume 5, Number 2, 126--130.

Abstract:
In this short note, a new approach is provided to prove that every nonzero continuous cosine function on a compact group $G$ is the normalized character of a representation of $G$ into the special unitary group $SU(2)$.

Topological games and strong quasi-continuity

Sun, 14 Aug 2011 19:03 EDT

Alireza Kamel Mirmostafaee

Source: Banach J. Math. Anal., Volume 5, Number 2, 131--137.

Abstract:
Let $X$ be a Baire space, $Y$ be a $W$-space and $Z$ be a regular topological space. We will show that every $KC$-function $f:X \times Y\to Z$ is strongly quasi-continuous at each point of $X \times Y$. In particular, when $X$ is a Baire space and $Y$ is Corson compact, every $KC$-function $f$ from $X \times Y$ to a Moore space $Z$ is jointly continuous on a dense subset of $X \times Y$. We also give a few applications of our results on continuity of group actions.

A glimpse at the Dunkl-Williams inequality

Sun, 14 Aug 2011 19:03 EDT

M. S. Moslehian, F. Dadipour, R. Rajic, A. Maric

Source: Banach J. Math. Anal., Volume 5, Number 2, 138--151.

Abstract:
In this paper we survey the results on the Dunkl-Williams inequality in normed linear spaces. These are related to the geometry of normed linear spaces, the characterizations of inner product spaces, some inequalities regarding operators on Hilbert spaces and elements of Hilbert $C^*$-modules.

Convex majorants method in the theory of nonlinear Volterra equations

Mon, 14 May 2012 12:57 EDT

Denis N. Sidorov , Nikolai A. Sidorov

Source: Banach J. Math. Anal., Volume 6, Number 1, 1--10.

Abstract:
The main solutions in the sense of Kantorovich of nonlinear Volterra operator-integral equations are constructed. Convergence of the successive approximation method is established through studies of the majorant integral equations and the majorant algebraic equations. Estimates are derived for the solutions and for the intervals on the right margin of which the solution of nonlinear Volterra operator-integral equation has blow-up or solution start branching.

Convergence theorems based on the shrinking projection method for hemi-relatively nonexpansive mappings, variational inequalities and equilibrium problems

Mon, 14 May 2012 12:57 EDT

Zi-Ming Wang, Mi Kwang Kang, Yeol Je Cho

Source: Banach J. Math. Anal., Volume 6, Number 1, 11--34.

Abstract:
In this paper, we introduce a new hybrid projection algorithm based on the shrinking projection methods for two hemi-relatively nonexpansive mappings. Using the new algorithm, we prove some strong convergence theorems for finding a common element in the fixed points set of two hemi-relatively nonexpansive mappings, the solutions set of a variational inequality and the solutions set of an equilibrium problem in a uniformly convex and uniformly smooth Banach space. Furthermore, we apply our results to finding zeros of maximal monotone operators. Our results extend and improve the recent ones announced by Li [J. Math. Anal. Appl. 295 (2004) 115--126], Fan [J. Math. Anal. Appl. 337 (2008) 1041--1047], Liu [J. Glob. Optim. 46 (2010) 319--329], Kamraksa and Wangkeeree [J. Appl. Math. Comput. DOI: 10.1007/s12190-010-0427-2] and many others.

Complete monotonicity of a function involving the ratio of gamma functions and applications

Mon, 14 May 2012 12:57 EDT

Feng Qi, Chun-Fu Wei, Bai-Ni Guo

Source: Banach J. Math. Anal., Volume 6, Number 1, 35--44.

Abstract:
In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result of Guo and Qi [Taiwanese J. Math. 7 (2003), no. 2, 239--247] and others. As applications, several inequalities involving the volume of the unit ball in $\mathbb{R}^n$ are derived, which refine, generalize and extend some known inequalities.

Linear maps preserving pseudospectrum and condition spectrum

Mon, 14 May 2012 12:57 EDT

G. Krishna Kumar, S. H. Kulkarni

Source: Banach J. Math. Anal., Volume 6, Number 1, 45--60.

Abstract:
We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its $\epsilon$-perturbation. Several results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let $A, B$ be complex unital Banach algebras and $\epsilon$ is positive. Let $\Phi: A\rightarrow B$ be an $\epsilon$-pseudospectrum preserving linear onto map. Then $\Phi$ preserves spectrum. If $A$ and $B$ are uniform algebras, then, $\Phi$ is an isometric isomorphism. (2) Let $A, B$ be uniform algebras and $\epsilon \in (0,1)$. Let $\Phi:A\rightarrow B$ be an $\epsilon$-condition spectrum preserving linear map. Then $\Phi$ is an $\epsilon^{'}$-almost multiplicative map, where $\epsilon, \epsilon^{'}$ tend to zero simultaneously.

Optimal range theorems for operators with p-th power factorable adjoints

Mon, 14 May 2012 12:57 EDT

Orlando Galdames Bravo Galdames Bravo, Enrique A. Sanchez Perez

Source: Banach J. Math. Anal., Volume 6, Number 1, 61--73.

Abstract:
Consider an operator $T:E \to X(\mu)$ from a Banach space $E$ to a Banach function space $X(\mu)$ over a finite measure $\mu$ such that its dual map is $p$-th power factorable. We compute the optimal range of $T$ that is defined to be the smallest Banach function space such that the range of $T$ lies in it and the restricted operator has $p$-th power factorable adjoint. For the case $p=1$, the requirement on $T$ is just continuity, so our results give in this case the optimal range for a continuous operator. We give examples from classical and harmonic analysis, as convolution operators, Hardy type operators and the Volterra operator.

Quadruple fixed point theorems for nonlinear contractions in partially ordered metric spaces

Mon, 14 May 2012 12:57 EDT

Erdal Karapinar, Vasile Berinde

Source: Banach J. Math. Anal., Volume 6, Number 1, 74--89.

Abstract:
In this paper we obtain existence and uniqueness results for quadruple fixed points of operators $F: X^4 \rightarrow X$. We also give some examples to support our results.

Almost automorphic solutions of hyperbolic evolution equations

Mon, 14 May 2012 12:57 EDT

Bruno de Andrade, Claudio Cuevas, Erwin Henriquez

Source: Banach J. Math. Anal., Volume 6, Number 1, 90--100.

Abstract:
In this work we deals with almost automorphic behavior of solutions of a class of semilinear evolution equations. To achieve our goal we use interpolation theory and fixed point theory. As application, we examine sufficient conditions for existence of almost automorphic solutions of equations of the heat conduction theory.

Common fixed point theorems for contractive mappings satisfying $\Phi$-maps in G-metric spaces

Mon, 14 May 2012 12:57 EDT

Anchalee Kaewcharoen

Source: Banach J. Math. Anal., Volume 6, Number 1, 101--111.

Abstract:
We prove the existence of the unique common fixed point theorems of a pair of weakly compatible mappings satisfying $\Phi$-maps in $G$-metric spaces. These results generalize the well-known results in the literature.

Composition operators from Nevanlinna type spaces to Bloch type spaces

Mon, 14 May 2012 12:57 EDT

Ajay K. Sharma, Sei-Ichiro Ueki

Source: Banach J. Math. Anal., Volume 6, Number 1, 112--123.

Abstract:
Let $X$ and $Y$ be complete metric spaces of analytic functions over the unit disk in the complex plane. A linear operator $T: X \to Y$ is a bounded operator with respect to metric balls if $T$ takes every metric ball in $X$ into a metric ball in $Y$. We also say that $T$ is metrically compact if it takes every metric ball in $X$ into a relatively compact subset in $Y$. In this paper we will consider these properties for composition operators from Nevanlinna type spaces to Bloch type spaces.

Polynomial functions and spectral synthesis on Abelian groups

Mon, 14 May 2012 12:57 EDT

Laszlo Szekelyhid

Source: Banach J. Math. Anal., Volume 6, Number 1, 124--131.

Abstract:
Spectral synthesis deals with the description of translation invariant function spaces. It turns out that the basic building blocks of this description are the exponential monomials, which are built up from exponential functions and polynomial functions. The author collaborated with Laczkovich [Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 1, 103--120] proved that spectral synthesis holds on an Abelian group if and only if the torsion free rank of the group is finite. The author [Aequationes Math. 70 (2005), no. 1-2, 122--130] showed that the torsion free rank of an Abelian group is strongly related to the properties of polynomial functions on the group. Here we show that spectral synthesis holds on an Abelian group if and only if the ring of polynomial functions on the group is Noetherian.

Comparison of one-sided modules

Mon, 14 May 2012 12:57 EDT

Kunal Mukherjee

Source: Banach J. Math. Anal., Volume 6, Number 1, 132--138.

Abstract:
Given an inclusion $N\subset M$ of $\rm{II}_{1}$ factors with trivial relative commutant, this paper lists all operators $x,y\in M$ such that the left $N$-module generated by $x$ is equal to or contained in the right $N$-module generated by $y$.

An extension of Ky Fan's dominance theorem

Mon, 14 May 2012 12:57 EDT

Rahim Alizadeh, Mohammad B. Asadi

Source: Banach J. Math. Anal., Volume 6, Number 1, 139--146.

Abstract:
We prove that for a separable Hilbert space $\mathcal{H}$ with an orthonormal basis $\{e_i\}_{i=1}^\infty$, the equality $\|\cdot\| =\|\sum_{i=1}^{\infty}s_i(\cdot)e_i\otimes e_i \|$ holds for all unitarily invariant norms on $\mathbb{B}(\mathcal{H})$ and Ky Fan's dominance theorem remains valid on $\mathbb{B}(\mathcal{H})$.

Bishop's property $(\beta)$ and Riesz Idempotent for k-quasi-paranormal operators

Mon, 14 May 2012 12:57 EDT

Salah Mecheri

Source: Banach J. Math. Anal., Volume 6, Number 1, 147--154.

Abstract:
The study of operators satisfying Bishop's property $(\beta)$ is of significant interest and is currently being done by a number of mathematicians around the world. Recently Uchiyama and Tanahashi [Oper. Matrices 4 (2009), 517--524] showed that a paranormal operator has Bishop's property $(\beta)$. In this paper we introduce a new class of operators which we call the class of $k$-quasi-paranormal operators. An operator $T$ is said to be a $k$-quasi-paranormal operator if it satisfies $||T^{k+1}x||^{2}\leq||T^{k+2}x|||T^{k}x||$ for all $x\in H$ where k is a natural number. This class of operators contains the class of paranormal operators and the class of quasi-class $A$ operators. We prove basic properties and give a structure theorem of $k$-quasi-paranormal operators. We also show that Bishop's property $(\beta)$ holds for this class of operators. Finally, we prove that if $E$ is the Riesz idempotent for a nonzero isolated point $\lambda_{0}$ of the spectrum of a $k$-quasi-paranormal operator $T$, then $E$ is self-adjoint if and only if the null space of $T-\lambda_{0},\, \ker(T-\lambda_{0})\subseteq \ker(T^{*}-\overline{\lambda_{0}})$.

Strong Arens irregularity of bilinear mappings and reflexivity

Mon, 14 May 2012 12:57 EDT

Ali Akbar Khadem-Maboudi, Hamid Reza Ebrahimi Vishki

Source: Banach J. Math. Anal., Volume 6, Number 1, 155--160.

Abstract:
We provide a sufficient condition for strong (Arens) irregularity of certain bounded bilinear maps, which applies in particular to the adjoint of Banach module actions. We then apply our result to improve several known results concerning to the relation between Arens regularity of certain Banach module actions and reflexivity.

$L^1$-convergence of greedy algorithm by generalized Walsh system

Mon, 14 May 2012 12:57 EDT

Sergo A. Episkoposian

Source: Banach J. Math. Anal., Volume 6, Number 1, 161--174.

Abstract:
In this paper we consider the generalized Walsh system and a problem $L^1$- convergence of greedy algorithm of functions after changing the values on small set.

Banach function algebras and certain polynomially norm-preserving maps

Fri, 13 Jul 2012 16:09 EDT

Maliheh Hosseini , Fereshteh Sady

Source: Banach J. Math. Anal., Volume 6, Number 2, 1--18.

Abstract:
Let $A$ and $B$ be Banach function algebras on compact Hausdorff spaces $X$ and $Y$, respectively. Given a non-zero scalar $\alpha$and $s,t\in \Bbb N$ we characterize the general form of suitable powers of surjective maps $T, T': A \longrightarrow B$ satisfying $\|(Tf)^s (T'g)^t-\alpha\|_Y=\|f^s g^t-\alpha \|_X$, for all $f,g \in A$, where $\|\cdot \|_X$ and $\|\cdot \|_Y$ denote the supremum norms on $X$ and $Y$, respectively. A similar result is given for the case where $T=T'$ and $T$ is defined between certain subsets of $A$ and $B$. We also show that if $T: A\longrightarrow B$ is a surjective map satisfying the stronger condition$R_\pi((Tf)^{s}(Tg)^{t}-\alpha)\cap R_\pi(f^{s}g^{t}-\alpha)\neq\varnothing $ for all $f,g \in A$, where $R_\pi(\cdot)$ denotes the peripheral range of the algebra elements, then there exists a homeomorphism $\varphi$ from the Choquet boundary $c(B)$ of $B$ onto the Choquet boundary $c(A)$ of $A$ such that $(Tf)^{d}(y)=(T1)^{d}(y)\,(f \circ \varphi(y))^{d}$ for all $f\in A$ and $y\in c(B)$,where $d$ is the greatest common divisor of $s$ and $t$.

On generalized ($m, n, l$)-Jordan centralizers of some algebras

Fri, 13 Jul 2012 16:09 EDT

Jiankui Li, Qihua Shen, Jianbin Guo

Source: Banach J. Math. Anal., Volume 6, Number 2, 19--37.

Abstract:
Let $\mathcal{A}$ be a unital algebra over a number field $\mathbb{K}$. A linear mapping $\delta$ from $\mathcal{A}$ into itself is called a generalized ($m, n, l$)-Jordan centralizer if it satisfies $(m+n+l)\delta(A^2)-m\delta(A)A-nA\delta(A)-lA\delta(I)A\in \mathbb{K}I$ for every $A\in \mathcal{A}$, where $m\geq0, n\geq0, l\geq0$ are fixed integers with $m+n+l\neq 0$. In this paper, we study generalized ($m, n, l$)-Jordan centralizers on generalized matrix algebras and some reflexive algebras alg$\mathcal{L}$, where $\mathcal{L}$ is a CSL or satisfies $\vee\{L: L\in \mathcal{J}(\mathcal{L})\}=X$ or $\wedge\{L_-: L\in \mathcal{J}(\mathcal{L})\}=(0)$, and prove that each generalized ($m, n, l$)-Jordan centralizer of these algebras is a centralizer when $m+l\geq1$ and $n+l\geq1$.

A Cuntz--Krieger uniqueness theorem for semigraph $C^*$-algebras

Fri, 13 Jul 2012 16:09 EDT

Bernhard Burgstaller

Source: Banach J. Math. Anal., Volume 6, Number 2, 38--57.

Abstract:
Higher rank semigraph algebras are introduced by mixing concepts of ultragraph algebras and higher rank graph algebras. This yields a kind of higher rank generalisation of ultragraph algebras. We prove Cuntz--Krieger uniqueness theorems for cancelling semigraph algebras and aperiodic saturated semigraph algebras.

Some new perturbation results for generalized inverses of closed linear operators in Banach spaces

Fri, 13 Jul 2012 16:09 EDT

Qianglian Huang, Lanping Zhu, Jiena Yu

Source: Banach J. Math. Anal., Volume 6, Number 2, 58--68.

Abstract:
We consider the perturbation and expression for the generalized inverse and Moore--Penrose inverse of closed linear operator under a weaker perturbation condition. As a application, we also investigate the perturbation for the Moore--Penrose inverse of closed $EP$ operator. Some new and interesting perturbation results and examples are obtained in this paper.

Subordination properties of multivalent functions defined by certain integral operator

Fri, 13 Jul 2012 16:09 EDT

Teodor Bulboaca, Mohamed K. Aouf, Rabha M. El-Ashwah

Source: Banach J. Math. Anal., Volume 6, Number 2, 69--85.

Abstract:
The object of this paper is to investigate some inclusion relationships and a number of other useful properties among certain subclasses of analytic and $p$-valent functions, which are defined here by certain integral operator.

Upper Beurling density of systems formed by translates of finite Sets of elements in $L^p(\mathbb{R}^d)$

Fri, 13 Jul 2012 16:09 EDT

Bei Liu, Rui Liu

Source: Banach J. Math. Anal., Volume 6, Number 2, 86--97.

Abstract:
In this paper, we prove that if a finite disjoint union of translates $\bigcup_{k=1}^n\{f_k(x-\gamma)\}_{\gamma\in\Gamma_k}$ in $L^p(\mathbb{R}^d)$ $(p \in (1, \infty))$ is a $p'$-Bessel sequence for some $p' \in (1, \infty)$, then the disjoint union $\Gamma=\bigcup_{k=1}^n \Gamma_k$ has finite upper Beurling density, and that if $\bigcup_{k=1}^n\{f_k(x-\gamma)\}_{\gamma\in\Gamma_k}$ is a $(C_q)$-system with $1/p+1/q=1$, then $\Gamma$ has infinite upper Beurling density. Thus, no finite disjoint union of translates in $L^p(\mathbb{R}^d)$ can form a $p'$-Bessel $(C_q)$-system for any $p'\in (1,\infty)$. Furthermore, by using techniques from the geometry of Banach spaces, we obtain that, for $p \in (1, \le2)$, no finite disjoint union of translates in $L^p(\mathbb{R}^d)$ can form an unconditional basis.

Traceability of positive integral operators in the absence of a metric

Fri, 13 Jul 2012 16:09 EDT

Mario H. de Castro, Valdir A. Menegatto, Ana P. Peron

Source: Banach J. Math. Anal., Volume 6, Number 2, 98--112.

Abstract:
We investigate the traceability of positive integral operators on $L^2(X,\mu)$ when $X$ is a Hausdorff locally compact second countable space and $\mu$ is a non-degenerate, $\sigma$-finite and locally finite Borel measure.\ This setting includes other cases proved in the literature, for instance the one in which $X$ is a compact metric space and $\mu$ is a special finite measure. The results apply to spheres, tori and other relevant subsets of the usual space $\mathbb{R}^m$.

Approximation by polynomials in rearrangement invariant quasi Banach function spaces

Fri, 13 Jul 2012 16:09 EDT

Ramazan Akgun

Source: Banach J. Math. Anal., Volume 6, Number 2, 113--131.

Abstract:
In the present work we deal with the approximation properties of certain linear polynomial operators in rearrangement invariant quasi Banach function spaces. We obtain some Jackson type direct theorem and sharp converse theorem of trigonometric approximation with respect to fractional positive order moduli of smoothness in these spaces.

Bounds for the ratio of two gamma functions---From Wendel's and related inequalities to logarithmically completely monotonic functions

Fri, 13 Jul 2012 16:09 EDT

Feng Qi, Qiu-Ming Luo

Source: Banach J. Math. Anal., Volume 6, Number 2, 132--158.

Abstract:
In the survey paper, along one of several main lines of bounding the ratio of two gamma functions, the authors retrospect and analyse Wendel's double inequality, Kazarinoff's refinement of Wallis' formula, Watson's monotonicity, Gautschi's double inequality, Kershaw's first double inequality, and the (logarithmically) complete monotonicity results of functions involving ratios of two gamma or $q$-gamma functions obtained by Bustoz, Ismail, Lorch, Muldoon, and other mathematicians.

A version of the Hermite--Hadamard inequality in a nonpositve curvature space

Fri, 13 Jul 2012 16:09 EDT

Cristian Conde

Source: Banach J. Math. Anal., Volume 6, Number 2, 159--167.

Abstract:
We obtain some Hermite--Hadamard type inequalities for convex functions in a global non-positive curvature space.

An interpolation theorem for sublinear operators on non-homogeneous metric measure spaces

Fri, 13 Jul 2012 16:09 EDT

Haibo Lin, Dongyong Yang

Source: Banach J. Math. Anal., Volume 6, Number 2, 168--179.

Abstract:
Let $({\mathcal X}, d, \mu)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors establish an interpolation result that a sublinear operator which is bounded from the Hardy space $H^1(\mu)$ to $L^{1,\,\infty}(\mu)$ and from $L^\infty(\mu)$ to the BMO-type space RBMO($\mu$) is also bounded on $L^p(\mu)$ for all $p\in(1,\,\infty)$. This extension is not completely straightforward and improves the existing result

Weighted classes of quaternion-valued functions

Fri, 13 Jul 2012 16:09 EDT

A. El-Sayed Ahmed, Saleh Omran

Source: Banach J. Math. Anal., Volume 6, Number 2, 180--191.

Abstract:
In this paper, we define the classes $F(p,q,s)$ of quaternion-valued functions, then we characterize quaternion Bloch functions by quaternion $F(p,q,s)$ functions in the unit ball of $\mathbb{R}^3$. Further, some important basic properties of these functions are also considered.

Square root for backward operator weighted shifts with multiplicity $2$

Fri, 13 Jul 2012 16:09 EDT

Bingzhe Hou, Geng Tian

Source: Banach J. Math. Anal., Volume 6, Number 2, 192--203.

Abstract:
As is well-known, each positive operator $T$ acting on a Hilbert space has a positive square root which is realized by means of functional calculus. However, it is not always true that an operator have a square root. In this paper, by means of Schauder basis theory we obtain that if a backward operator weighted shift $T$ with multiplicity $2$ is not strongly irreducible, then there exists a backward shift operator $B$ (maybe unbounded) such that $T=B^2$. Furthermore, the backward operator weighted shifts in the sense of Cowen-Douglas are also considered.

Operator inequalities and normal operators

Fri, 13 Jul 2012 16:09 EDT

Safa Menkad, Ameur Seddik

Source: Banach J. Math. Anal., Volume 6, Number 2, 204--210.

Abstract:
In the present paper, taking some advantages offered by the context of finite dimensional Hilbert spaces, we shall give a complete characterizations of certain distinguished classes of operators (self-adjoint, unitary reflection, normal) in terms of operator inequalities. These results extend previous characterizations obtained by the second author.

The refined Sobolev scale, interpolation and elliptic problems

Fri, 13 Jul 2012 16:09 EDT

Vladimir A. Mikhailets , Aleksandr A. Murach

Source: Banach J. Math. Anal., Volume 6, Number 2, 211--281.

Abstract:
The paper gives a detailed survey of recent results on elliptic problems in Hilbert spaces of generalized smoothness. The latter are the isotropic Hörmander spaces $H^{s,\varphi}:=B_{2,\mu}$, with $\mu(\xi)=\langle\xi\rangle^{s}\varphi(\langle\xi\rangle)$ for $\xi\in\mathbb{R}^{n}$. They are parametrized by both the real number $s$ and the positive function $\varphi$ varying slowly at $+\infty$ in the Karamata sense. These spaces form the refined Sobolev scale, which is much finer than the Sobolev scale $\{H^{s}\}\equiv\{H^{s,1}\}$ and is closed with respect to the interpolation with a function parameter. The Fredholm property of elliptic operators and elliptic boundary-value problems is preserved for this new scale. Theorems of various type about a solvability of elliptic problems are given. A~local refined smoothness is investigated for solutions to elliptic equations. New sufficient conditions for the solutions to have continuous derivatives are found. Some applications to the spectral theory of elliptic operators are given.

Table of Contents, Banach J. Math. Anal., vol. 7, no. 2 (2013)

Wed, 20 Mar 2013 08:57 EDT

Source: Banach J. Math. Anal., Volume 7, Number 2, --.

Matrix transformations and sequence spaces equations

Wed, 20 Mar 2013 08:57 EDT

Bruno de Malafosse , Vladimir Rakocevic

Source: Banach J. Math. Anal., Volume 7, Number 2, 1--14.

Abstract:
In this paper we study sequence spaces equations (SSE) with operators, which are determined by an identity whose each term is a sum or a sum of products of sets of the form $\chi _{a}\left( T\right)$ and $\chi _{f\left( x\right) }\left( T\right)$ where $f$ maps $U^{+}$ to itself, $\chi $ is either of the symbols $s$, $s^{0}$, or $s^{\left( c\right) }$. Then we solve five (SSE) of the form $\chi _{a}+\chi _{x}^{\prime }=\chi _{b}^{\prime }$, where $\chi $, $\chi ^{\prime }$ are either $s^{{{}0}}$, $s^{\left( c\right) }$, or $s$. We apply the previous results to the solvability of the systems $s_{a}^{0}+s_{x}\left( \Delta \right) =s_{b}$, $s_{x}\supset s_{b}$ and $s_{a}+s_{x}^{\left( c\right) }\left( \Delta \right) =s_{b}^{\left( c\right) }$, $s_{x}^{\left( c\right) }\supset s_{b}^{\left( c\right) }$. Finally we solve the (SSE) with operators defined by $\chi _{a}\left( C\left( \lambda \right) D_{\tau }\right) +s_{x}^{\left( c\right) }\left( C\left( \mu \right) D_{\tau }\right) =s_{b}^{\left( c\right) }$ where $\chi $ is either $s^{0}$, or $s$.

Refinements and reverses of means inequalities for Hilbert space operators

Wed, 20 Mar 2013 08:57 EDT

Omar Hirzallah , Fuad Kittaneh, Mario Krnic, Neda Lovricevic, Josip Pecaric

Source: Banach J. Math. Anal., Volume 7, Number 2, 15 --29 .

Abstract:
In this paper we derive some improvements of means inequalities for Hilbert space operators. More precisely, we obtain refinements and reverses of the arithmetic-geometric operator mean inequality. As an application, we also deduce an improved variant for the refined arithmetic--Heinz mean inequality. We also present some eigenvalue inequalities for differences of certain operator means.

Weyl type theorem and spectrum for $(p,k)$-quasiposinormal operators

Wed, 20 Mar 2013 08:57 EDT

D. Senthilkumar, D. Kiruthika , P. Maheswari Naik

Source: Banach J. Math. Anal., Volume 7, Number 2, 30 --41 .

Abstract:
Let $T$ be a $(p,k)$-quasiposinormal operator on a complex Hilbert space $\mathcal{H}$, i.e $T^{*k}(c^{2}(T^{*} T)^{p}-(T T^{*})^{p})T^{k} \geq 0$ for a positive integer $p \in (0,1]$, some $c > 0$ and a positive integer $k$. In this paper, we prove that the spectral mapping theorem for Weyl spectrum holds for $(p, k)$ - quasiposinormal operators. We show that the Weyl type theorems holds for $(p,k)$- quasiposinormal. We prove that if $T^{*}$ is $(p,k)$-quasiposinormal, then generalized $a$-Weyl's theorem holds for $T$. Also we prove that $\sigma_{jp}(T)-\{0\} = \sigma_{ap}(T)-\{0\}$ holds for $(p,k)$-quasiposinormal operator.

Pseudo Asymptotic Solutions of Fractional Order Semilinear Equations

Wed, 20 Mar 2013 08:57 EDT

Edgardo Alvarez-Pardo , Carlos Lizama

Source: Banach J. Math. Anal., Volume 7, Number 2, 42 --52 .

Abstract:
Using a generalization of the semigroup theory of linear operators, we prove existence and uniqueness of mild solutions for the semilinear fractional order differential equation $${D}^{\alpha+1}_t u(t) + \mu {D}_t^{\beta} u(t) - Au(t) = f(t,u(t)), t\in (0,\infty), \alpha \in (0,\infty), \alpha \leq \beta \leq 1, \, \mu \geq 0, $$ with the property that the solution can be written as $u=f+h$ where $f$ belongs to the space of periodic (resp. almost periodic, compact almost automorphic, almost automorphic) functions and $h$ belongs to the space $ P_0(\mathbb{R}_{+},X):= \{ \phi\in BC(\mathbb{R}_{+},X) \, :\,\, \lim_{T \to \infty}\frac{1}{T} \int_{0}^{T}||\phi(s)||ds=0 \}$. Moreover, this decomposition is unique.

Weak ergodicity of nonhomogeneous Markov chains on noncommutative $L^1$-spaces

Wed, 20 Mar 2013 08:57 EDT

Farrukh Mukhamedov

Source: Banach J. Math. Anal., Volume 7, Number 2, 53 --73.

Abstract:
In this paper we study certain properties of Dobrushin's ergodicity coefficient for stochastic operators defined on noncommutative $L^1$-spaces associated with semi-finite von Neumann algebras. Such results extends the well-known classical ones to a noncommutative setting. This allows us to investigate the weak ergodicity of nonhomogeneous discrete Markov processes (NDMP) by means of the ergodicity coefficient. We provide a sufficient conditions for such processes to satisfy the weak ergodicity. Moreover, a necessary and sufficient condition is given for the satisfaction of the $L^1$-weak ergodicity of NDMP. It is also provided an example showing that $L^1$-weak ergodicity is weaker that weak ergodicity. We applied the main results to several concrete examples of noncommutative NDMP.

Special operator classes and their properties

Wed, 20 Mar 2013 08:57 EDT

Mubariz Tapdigoglu Karaev , Mehmet Gurdal, Ulas Yamanci

Source: Banach J. Math. Anal., Volume 7, Number 2, 74 --85 .

Abstract:
We introduce some special operator classes and study in terms of Berezin symbols their properties. In particular, we give some characterizations of compact operators and Schatten-von Neumann class operators in terms of Berezin symbols. We also consider some classes of compact operators on a Hilbert space $H,$ which are generalizations of the well known Schatten-von Neumann classes of compact operators. Namely, for any number $p \in (0,\infty)$ and the sequence $w:=(w_{n})_{n\geq0}$ of complex numbers $w_{n},$ $n\geq 0,$ we define the following classes of compact operators on $H $: $$S_{p}^{w}(H)=\left\{ K\in S_{\infty}(H):\sum_{n=0}^{\infty}(s_{n} (K))^{p}w_{n}^{p}\hbox{ is convergent series }\right\}, $$ where $s_{n}(K)$ denotes the $n$th singular number of the operator $K$. The characterizations of these classes are given in terms of Berezin symbols.

On the boundedness and compactness of a certain integral operator

Wed, 20 Mar 2013 08:57 EDT

S. M. Farsani

Source: Banach J. Math. Anal., Volume 7, Number 2, 86 --102 .

Abstract:
Let $\alpha \in (0, \infty)$ and $\beta \in (1, \infty)$. In the present work, the necessary and sufficient conditions for the boundedness and compactness of the integral operator of the form \begin{equation*} L_{\alpha, \beta} f(x):=v(x)\int_0^x \frac{\ln^{\beta-1}(\frac{x}{y})f(y)u(y)dy}{(x-y)^{1-\alpha}} ,\,\,\,\, x>0, \end{equation*} from $L^p\to L^q$, with locally integrable non-negative weight functions $u$ and $v$, in the case $p,q \in (0, \infty), p>\max(1/{\alpha},1),$ provided $u$ is non-increasing on $\mathbb{R}^+:=[0,\infty)$ are found.

Noncommutative spectral synthesis for the involutive Banach algebra associated with a topological dynamical system

Wed, 20 Mar 2013 08:57 EDT

Marcel de Jeu , Jun Tomiyama

Source: Banach J. Math. Anal., Volume 7, Number 2, 103 --135 .

Abstract:
If $\Sigma=(X,\sigma)$ is a topological dynamical system, where $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then a crossed product involutive Banach algebra $\ell^1$ is naturally associated with these data. If $X$ consists of one point, then $\ell^1$ is the group algebra of the integers, hence the general$\ell^1$could be regarded as a noncommutative $\ell^1$-algebra. In this paper, we study spectral synthesis for the closed ideals of $\ell^1$ in two versions, one modeled after $C(X)$and one modeled after $\ell^1(\mathbb{Z})$. We identify the closed ideals which are equal to (what is the analogue of) the kernel of their hull, and determine when this holds for all closed ideals, i.e., when spectral synthesis holds. In both models, this is the case precisely when $\Sigma$ is free.

Algebraically paranormal operators\\ on Banach spaces

Wed, 20 Mar 2013 08:57 EDT

Pietro Aiena

Source: Banach J. Math. Anal., Volume 7, Number 2, 136 --145 .

Abstract:
In this paper we show that a bounded linear operator on a Banach space $X$ is polaroid if and only if $p(T)$ is polaroid for some polynomial $p$. Consequently, algebraically paranormal operators defined on Banach spaces are hereditarily polaroid. Weyl type theorems are also established for perturbations $f(T+K)$, where $T$ is algebraically paranormal, $K$ is algebraic and commutes with $T$, and $f$ is an analytic function, defined on an open neighborhood of the spectrum of $T+K$, such that $f$ is nonconstant on each of the components of its domain. These results subsume recent results in this area.

$(X_{d}, X_{d}^{*})$-Bessel multipliers in Banach spaces

Wed, 20 Mar 2013 08:57 EDT

Mohammad Hasan Faroughi , Elnaz Osgooei , Asghar Rahimi

Source: Banach J. Math. Anal., Volume 7, Number 2, 146 --161 .

Abstract:
Multipliers have recently been introduced as operators for Bessel sequences and frames in Hilbert spaces. In this paper, we define the concept of $(X_{d}, X_{d}^{*})$ and $(l^{\infty}, X_{d}, X_{d}^{*})$-Bessel multipliers in Banach spaces and investigate the compactness of these multipliers. Also, we study the possibility of invertibility of $(l^{\infty}, X_{d}, X_{d}^{*})$-Bessel multiplier depending on the properties of its corresponding sequences and its symbol. Furthermore, we prove that every $(X_{d}, X_{d}^{*})$-Bessel multiplier is a $\lambda$-nuclear operator.

Matrix inequalities related to Hölder inequality

Wed, 20 Mar 2013 08:57 EDT

Hussien Albadawi

Source: Banach J. Math. Anal., Volume 7, Number 2, 162 --171 .

Abstract:
Matrix inequalities of Hölder type are obtained. Among other inequalities, it is shown that if $p,q \in (2,\infty) $ and $r>1$ with $1/p+1/q=1-1/r$, then for any $A_{i},B_{i}\in M_{n}\left(\mathbb{C} \right) $ and $\alpha _{i}\in \left[ 0,1\right] $ $\left( i=1,2,\cdots ,m\right) $ with $\sum\limits_{i=1}^{m}\alpha _{i}=1$, we have% \begin{equation*} \left\vert \sum\limits_{i=1}^{m}\alpha _{i}^{1/r}B_{i}A_{i}\right\vert \leq \left( \sum\limits_{i=1}^{m}\left\vert A_{i}\right\vert ^{p}\right) ^{1/p} \end{equation*}% whenever $\sum\limits_{i=1}^{m}\left\vert B_{i}^{\ast }\right\vert ^{q}\leq I $. Related unitarily invariant norm inequalities are also presented.

Frames and Riesz bases for Banach spaces, and Banach spaces of vector-valued sequences

Wed, 20 Mar 2013 08:57 EDT

Kyugeun Cho, Ju Myung Kim, Han Ju Lee

Source: Banach J. Math. Anal., Volume 7, Number 2, 172 --193 .

Abstract:
This paper is devoted to an investigation of frames and Riesz bases for general Banach sequence spaces. We establish various relationships between Bessel (respectively, frames) and Riesz sequences (respectively, Riesz bases), and then some of their applications are presented. Some recent results for Banach frames and atomic decompositions are sharpened with simple proofs. Banach spaces consisting of Bessel or Riesz sequences are introduced and it is shown that they are isometrically isomorphic to some Banach spaces of bounded linear operators, and that some subspaces of those Banach spaces are isometrically isomorphic to some Banach spaces of compact operators.

On selfadjoint dilation of the dissipative extension of a direct sum differential operator

Wed, 20 Mar 2013 08:57 EDT

Ekin Ugurlu , Bilender P. Allahverdiev

Source: Banach J. Math. Anal., Volume 7, Number 2, 194--207 .

Abstract:
In this paper, we describe all maximal dissipative, maximal accretive and selfadjoint extensions of the minimal symmetric direct sum differential operators. Further using the equivalence of the Lax-Phillips scattering function and the Sz.-Nagy-Foia\c{s} characteristic function we show that all eigen and associated functions of the maximal dissipative extension of the minimal symmetric direct sum operator are complete in $L_{w}^{2}(\Omega ),$ where $\Omega =\Omega _{1}\cup \Omega _{2},$ $\Omega _{1}=(0,c)$ and $\Omega _{2}=(c,\infty ).$

The solvability and the exact solution of a system of real quaternion matrix equations

Wed, 20 Mar 2013 08:57 EDT

Xiang Zhang , Qing-Wen Wang

Source: Banach J. Math. Anal., Volume 7, Number 2, 208 --224 .

Abstract:
In this paper, we establish necessary and sufficient conditions for the solvability of the system of real quaternion matrix equations $$ \left\{ \begin{array} {l}{A}_{1}X=C_{1},~ \\ YB_{1}=D_{1}, \\ A_{2}Z=C_{2},ZB_{2}=D_{2},A_{3}ZB_{3}=C_{3}, \\ A_{4}X+YB_{4}+C_{4}ZD_{4}=E_{1}. \end{array}\right. $$ We also present an expression of the general solution to the system. The findings of this paper widely extend the known results in the literature.

On high dimensional maximal operators

Wed, 20 Mar 2013 08:57 EDT

J. M. Aldaz , J. Perez Lazaro

Source: Banach J. Math. Anal., Volume 7, Number 2, 225 --243 .

Abstract:
In this note we describe some recent advances in the area of maximal function inequalities. We also study the behaviour of the centered Hardy--Littlewood maximal operator associated to certain families of doubling, radial decreasing measures, and acting on radial functions. In fact, we precisely determine when the weak type $(1,1)$ bounds are uniform in the dimension.