Showing papers in "Banach Journal of Mathematical Analysis in 2012"
TL;DR: In this article, the authors retrospect and analyse Wen- del's double inequality, Kazarino's renement of Wallis' formula, Watson's monotonicity, Gautschi's double inequalities, Kershaw's rst double inequality and the (logarithmically) complete monotonic results of functions involving ratios of two gamma or q-gamma functions obtained by Bustoz, Ismail, Lorch, Muldoon, and other mathematicians.
Abstract: In the survey paper, along one of several main lines of bounding the ratio of two gamma functions, the authors retrospect and analyse Wen- del's double inequality, Kazarino's renement of Wallis' formula, Watson's monotonicity, Gautschi's double inequality, Kershaw's rst 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.
88 citations
TL;DR: In this article, the existence and uniqueness results for quadruple-ple fixed points of operators F : X 4! X were obtained and some examples to support their results were given.
Abstract: In this paper we obtain existence and uniqueness results for quadru- ple fixed points of operators F : X 4 ! X. We also give some examples to support our results.
70 citations
TL;DR: A detailed survey of recent results on elliptic problems in Hilbert spaces of generalized smoothness can be found in this article, where a local refined smoothness is investigated for solutions to elliptic equations and sufficient conditions for the solutions to have continuous derivatives are found.
Abstract: The paper gives a detailed survey of recent results on elliptic problems in Hilbert spaces of generalized smoothness. The latter are the isotropic Hormander spaces H := B2,μ, with μ(ξ) = 〈ξ〉φ(〈ξ〉) for ξ ∈ R. They are parametrized by both the real number s and the positive function φ varying slowly at +∞ in the Karamata sense. These spaces form the refined Sobolev scale, which is much finer than the Sobolev scale {H} ≡ {H} 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.
58 citations
TL;DR: In this article, necessary and sucient conditions are presented for a function involving a ratio of gamma functions to be logarithmically com- pletely monotonic.
Abstract: In the paper, necessary and sucient conditions are presented for a function involving a ratio of gamma functions to be logarithmically com- pletely 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 R n are derived, which refine, generalize and extend some known inequalities.
39 citations
TL;DR: In this article, the main solutions in the sense of Kantorovich of nonlinear Volterra operator-integral equations are constructed, and the convergence of the successive ap- proximation method is established through studies of the majorant integral equations and majorant algebraic equations.
Abstract: The main solutions in the sense of Kantorovich of nonlinear Volterra operator-integral equations are constructed. Convergence of the successive ap- proximation 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.
29 citations
TL;DR: The class of k-quasi-paranormal operators was introduced in this article, which is the class of operators that satisfy the Bishop's property for all x 2 H where k is a natural number.
Abstract: The study of operators satisfying Bishop's property ( ) is of sig- nificant 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 ( ). 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 || T k+2 x|||T k x|| for all x 2 H where k is a natural number. This class of operators contains the class of paranormal oper- ators 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 ( ) holds for this class of operators. Finally, we prove that if E is the Riesz idempotent for a nonzero isolated point 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 0, ker(T 0) ker(T 0).
28 citations
TL;DR: In this paper, the existence of common fixed point theorems of a pair of weakly compatible mappings satisfying $G$-maps in metric spaces has been proved, and the results generalize the well-known results in the literature.
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.
25 citations
TL;DR: In this paper, properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its -perturbation are discussed and results about linear maps preserving pseudosensorrum/ condition spectrum are proved.
Abstract: We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its -perturbation. Sev- eral results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let A,B be complex unital Banach algebras and
25 citations
TL;DR: In this article, the authors established an interpolation result that a sublinear operator which is bounded from the Hardy space $H^1(\mu)$ to $L^{1,\,\infty}(\mu), and from $L^\in-to-the-BMO-type space RBMO($\mu$) is also bounded on $Lp(1, \, ∞) for all
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
22 citations
TL;DR: In this article, the classes of quaternion-valued functions are defined, and the properties of these functions are investigated, and some important basic properties of the functions are also considered.
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.
17 citations
TL;DR: In this article, the authors consider the properties of composition operators from Nevanlinna type spaces to Bloch type spaces and show that a linear operator T : X! Y is a bounded operator with respect to metric balls if T takes every metric ball in X into a relatively compact subset in Y.
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 ! 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.
TL;DR: In this paper, a new hybrid projection algorithm based on the shrinking projection method for two hemi-relatively nonexpan-sive mappings is introduced, and strong convergence theorems for finding a common element in the fixed points set of two relatively nonexpansive mapping mappings, the solution set of a variational inequality and the solutions set of an equilibrium problem in a uniformly convex and uniformly smooth Banach space are established.
Abstract: In this paper, we introduce a new hybrid projection algorithm based on the shrinking projection methods for two hemi-relatively nonexpan- sive 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: 101007/s12190-010-0427-2) and many others
TL;DR: In this paper, the perturbation and expression for generalized inverse and Moore-Penrose inverse of closed linear operator under a weaker perturbant condition were investigated. But the perturbed expression for the generalized inverse was not considered.
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 interest- ing perturbation results and examples are obtained in this paper.
TL;DR: In this paper, the Hermite-Hadamard type inequalities for convex functions in a global non-positive curvature space were obtained for a convex convex function.
Abstract: We obtain some Hermite{Hadamard type inequalities for convex functions in a global non-positive curvature space.
TL;DR: In this paper, a sufficient condition for strong (Arens) irregularity of bounded bilinear maps is provided, which applies in particular to the adjoint of Banach module actions.
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.
TL;DR: In this article, the optimal range of a continuous operator is defined as the smallest Banach function space such that the range of T lies in it and the restricted operator has p-th power factorable adjoint.
Abstract: Consider an operator T : E ! X(µ) from a Banach space E to a Banach function space X(µ) over a finite measure µ 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.
TL;DR: In this article, the generalized Walsh system and a problem of convergence of greedy algorithm of functions after changing the values on small set were considered and the convergence was shown to be NP-hard.
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.
TL;DR: In this paper, the authors investigate some inclusion rela- tionships and a number of other useful properties among certain subclasses of analytic and p-valent functions, which are dened here by certain integral operator.
Abstract: The object of this paper is to investigate some inclusion rela- tionships and a number of other useful properties among certain subclasses of analytic and p-valent functions, which are dened here by certain integral operator.
TL;DR: For a separable Hilbert space H with an orthonor-mal basis {ei} 1=1, the equality k·k = k P 1 = 1 si(·)ei eik holds for all unitarily invariant norms on H and Ky Fan's dominance theorem remains valid on H as discussed by the authors.
Abstract: We prove that for a separable Hilbert space H with an orthonor- mal basis {ei} 1=1, the equality k·k = k P 1=1 si(·)ei eik holds for all unitarily invariant norms on B(H) and Ky Fan's dominance theorem remains valid on B(H).
TL;DR: A generalized (m, n, l)-Jordan centralizer is defined in this paper, where a linear mapping from A into itself is called a generalized Jordan centralizer if it satises.
Abstract: LetA be a unital algebra over a number eld K. A linear mapping from A into itself is called a generalized (m, n, l)-Jordan centralizer if it satises (
TL;DR: In this article, the authors deal with almost automorphic behavior of so-lutions of a class of semilinear evolution equations using interpolation theory and fixed point theory.
Abstract: In this work we deals with almost automorphic behavior of so- lutions of a class of semilinear evolution equations. To achieve our goal we use interpolation theory and fixed point theory. As application, we examine sucient conditions for existence of almost automorphic solutions of equations of the heat conduction theory.
TL;DR: In this paper, it was shown that no finite disjoint union of translates in $L^p(mathbb{R}^d)$ can form a Bessel system with constant upper Beurling density.
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.
TL;DR: In this paper, the approximation properties of certain linear polynomial operators in rearrangement invariant quasi Ba-nach function spaces were studied. And they obtained some Jackson type direct theorem and sharp converse theorem of trigonometric approximation with respect to fractional positive order moduli of smoothness in these spaces.
Abstract: In the present work we deal with the approximation properties of certain linear polynomial operators in rearrangement invariant quasi Ba- nach 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.
TL;DR: In this article, the traceability of positive integral operators on L 2 (X; ) is investigated in the setting where X is a Hausdor locally compact second countable space and is a non-degenerate, -nite and locally nite Borel measure.
Abstract: We investigate the traceability of positive integral operators on L 2 (X; ) when X is a Hausdor locally compact second countable space and is a non-degenerate, -nite and locally nite Borel measure. This setting includes other cases proved in the literature, for instance the one in which X is a compact metric space and is a special nite measure. The results apply to spheres, tori and other relevant subsets of the usual space R m .
TL;DR: In this article, Cuntz{ Krieger uniqueness theorems for cancelling semigraph algebras and aperiodic saturated semigraphal algesbras were proved.
Abstract: Higher rank semigraph algebras are introduced by mixing con- cepts 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.
TL;DR: In this article, the authors characterize the general form of suitable powers of surjective maps for Banach function algebras on compact Hausdor spaces X and Y, respectively.
Abstract: Let A and B be Banach function algebras on compact Hausdor spaces X and Y , respectively. Given a non-zero scalar and s;t 2 N we characterize the general form of suitable powers of surjective mapsT;T 0 : A! B satisfyingk(Tf) s (T 0 g) t kY =kf s g t kX, for all f;g2 A, wherek k X andk k Y denote the supremum norms on X and Y , respectively. A similar result is given for the case where T = T 0 and T is dened between certain subsets of A and B. We also show that if T : A ! B is a surjective map satisfying the stronger conditionR ((Tf) s (Tg) t )\R (f s g t )6 ? for all f;g2 A, where R ( ) denotes the peripheral range of the algebra elements, then there exists a homeomorphism ' from the Choquet boundary c(B) of B onto the Choquet boundaryc(A) ofA such that (Tf) d (y) = (T 1) d (y) (f '(y)) d for all f2 A and y2 c(B),where d is the greatest common divisor of s and t.
TL;DR: In this article, by means of Schauder basis theory, it is shown that if a backward operator weighted shift with multiplicity 2 is not strongly irreducible, then there exists a backward shift operator B$ (possibly unbounded) such that B = B^2.
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.
TL;DR: In this paper, the authors give a complete characteriza- tions of certain distinguished classes of operators (self-adjoint, unitary reec- tion, normal) in terms of operator inequalities.
Abstract: In the present paper, taking some advantages oered by the con- text of nite dimensional Hilbert spaces, we shall give a complete characteriza- tions of certain distinguished classes of operators (self-adjoint, unitary reec- tion, normal) in terms of operator inequalities. These results extend previous characterizations obtained by the second author.
TL;DR: In this article, the authors presented a list of operators x,y 2 M such that the left N-module generated by x is equal to or contained in the right one generated by y.
Abstract: Given an inclusion N M of II1 factors with trivial relative commutant, this paper lists all operators x,y 2 M such that the left N-module generated by x is equal to or contained in the right N-module generated by y
TL;DR: In this paper, it was shown that spectral synthesis holds on an Abelian group if and only if the ring of polynomial functions on the group is Noetherian and the torsion free rank of the group has no finite rank.
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.