Copies of c0(Г) and ?∞(Г)/c0(Г) in quotients of Banach spaces with applications to Orlicz and Marcinkiewicz spaces

Let X be a Banach space, let Y be its subspace, and let Г be an infinite set. We study the consequences of the assumption that an operator T embeds ?_221E;(Г) into X isomorphically with T(c₀(Г)) ⊂ Y. Under additional assumptions on T we prove the existence of isomorphic copies of c₀(Г^ℵ0) in X/Y, and complemented copies ?_∞(Г) in X/Y. In concrete cases we obtain a new information about the structure of X/Y. In particular, L∞[O,1]/C[O,1] contains a complemented copy of ?_∞/c₀, and some natural (lattice) quotients of real Orlicz and Marcinkiewicz spaces contain lattice-isometric and positively I-complemented copies of(real) ?_∞/c₀.     

On operator valued sequences of multipliers and R-boundedness

In recent papers (cf. [J.L. Arregui, O. Blasco, (p,q)-Summing sequences, J. Math. Anal. Appl. 274 (2002) 812-827; J.L. Arregui, O. Blasco, (p,q)-Summing sequences of operators, Quaest. Math. 26 (2003) 441-452; S. Aywa, J.H. Fourie, On summing multipliers and applications, J. Math. Anal. Appl. 253 (2001) 166-186; J.H. Fourie, I. Röntgen, Banach space sequences and projective tensor products, J. Math. Anal. Appl. 277 (2) (2003) 629-644]) the concept of (p,q)-summing multiplier was considered in both general and special context. It has been shown that some geometric properties of Banach spaces and some classical theorems can be described using spaces of (p,q)-summing multipliers. The present paper is a continuation of this study, whereby multiplier spaces for some classical Banach spaces are considered. The scope of this research is also broadened, by studying other classes of summing multipliers. Let E(X) and F(Y) be two Banach spaces whose elements are sequences of vectors in X and Y, respectively, and which contain the spaces c₀₀(X) and c₀₀(Y) of all X-valued and Y-valued sequences which are eventually zero, respectively. Generally spoken, a sequence of bounded linear operators (un)⊂L(X,Y) is called a multiplier sequence from E(X) to F(Y) if the linear operator from c₀₀(X) into c₀₀(Y) which maps (xi)∈c₀₀(X) onto (unxn)∈c₀₀(Y) is bounded with respect to the norms on E(X) and F(Y), respectively. Several cases where E(X) and F(Y) are different (classical) spaces of sequences, including, for instance, the spaces Rad(X) of almost unconditionally summable sequences in X, are considered. Several examples, properties and relations among spaces of summing multipliers are discussed. Important concepts like R-bounded, semi-R-bounded and weak-R-bounded from recent papers are also considered in this context.     

The characterization of compact operators on spaces of strongly summable and bounded sequences

We use the characterizations of the classes of all infinite matrices that map the spaces of sequences which are strongly summable or bounded by the Cesàro method of order 1 into the spaces of null or convergent sequences given by Ba?ar, Malkowsky and Altay [Matrix transformations on the matrix domains of triangles in the spaces of strongly C₁-summable and bounded sequences, Publ. Math. Debrecen 73 (1-2) (2008), 193-213] and the Hausdorff measure of noncompactness to characterize the classes of all compact operators between those spaces.     

Weighted composition operators between weighted spaces of vector-valued holomorphic functions on Banach spaces

Let B(E, F) be the Banach Space of all continuous linear operators from a Banach Space E into a Banach space F. Let U_X and U_Y be balanced open subsets of Banach spaces X and Y, respectively. Let V and W be two Nachbin families of continuous weights on U_X and U_Y, respectively. Let HV(U_X, E) (or HV₀(U_X, E)) and HW(U_Y, F) (or HW₀(U_Y, F)) be the weighted spaces of vector-valued holomorphic functions. In this paper, we investigate the holomorphic mappings ? : U_Y → U_X and ψ : U_Y → B(E, F) which generate weighted composition operators between these weighted spaces.     

Some properties and applications of weakly equicompact sets

Let X and Y be Banach spaces. A set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x _n) in X, there exists a subsequence (x _k(n)) so that (Tx_k(n)) is uniformly weakly convergent for T ∈ M. In this paper, the notion of weakly equicompact set is used to obtain characterizations of spaces X such that X ↩̸ ℓ₁, of spaces X such that B _X* is weak* sequentially compact and also to obtain several results concerning to the weak operator and the strong operator topologies. As another application of weak equicompactness, we conclude a characterization of relatively compact sets in when this space is endowed with the topology of uniform convergence on the class of all weakly null sequences. Finally, we show that similar arguments can be applied to the study of uniformly completely continuous sets. Received: 5 July 2006     

Asymptotically isometric copies of c0 and l in certain Banach spaces

If X is a separable Banach space, then X∗ contains an asymptotically isometric copy of l¹ if and only if there exists a quotient space of X which is asymptotically isometric to c₀. If X is an infinite-dimensional normed linear space and Y is any Banach space containing an asymptotically isometric copy of c₀, then L(X,Y) contains an isometric copy of l^∞. If X and Y are two infinite-dimensional Banach spaces and Y contains an asymptotically isometric copy of c₀, then contains a complemented asymptotically isometric copy of c₀.     

Matrix transformations on some sequence spaces related to strong Cesàro summability and boundedness

The spaces and introduced by Ayd?n and Ba?ar [C. Ayd?n, F. Ba?ar, Some new difference sequence spaces, Appl. Math. Comput. 157 (3) (2004) 677-693] can be considered as the matrix domains of a triangle in the sets of all sequences that are summable to zero, summable, and bounded by the Cesàro method of order 1. Here we define the sets of sequences which are the matrix domains of that triangle in the sets of all sequences that are summable, summable to zero, or bounded by the strong Cesàro method of order 1 with index p?1. We determine the β-duals of the new spaces and characterize matrix transformations on them into the sets of bounded, convergent and null sequences.     

The best generalized inverse of the linear operator in normed linear space

Let X,Y be normed linear spaces, T∈L(X,Y) be a bounded linear operator from X to Y. One wants to solve the linear problem Ax=y for x (given y∈Y), as well as one can. When A is invertible, the unique solution is x=A^-1y. If this is not the case, one seeks an approximate solution of the form x=By, where B is an operator from Y to X. Such B is called a generalised inverse of A. Unfortunately, in general normed linear spaces, such an approximate solution depends nonlinearly on y. We introduce the concept of bounded quasi-linear generalised inverse T^h of T, which contains the single-valued metric generalised inverse T^M and the continuous linear projector generalised inverse T⁺. If X and Y are reflexive, we prove that the set of all bounded quasi-linear generalised inverses of T, denoted by G_H(T), is not empty In the normed linear space of all bounded homogeneous operators, the best bounded quasi-linear generalised inverse T^h of T is just the Moore-Penrose metric generalised inverse T^M. In the case, X and Y are finite dimension spaces Rⁿ and R^m, respectively, the results deduce the main result by G.R. Goldstein and J.A. Goldstein in 2000.     

Operators on the space of vector-valued totally measurable functions

Let L(X,Y) stand for the space of all bounded linear operators between real Banach spaces X and Y, and let Σ be a σ-algebra of sets. A bounded linear operator T from the Banach space B(Σ,X) of X-valued Σ-totally measurable functions to Y is said to be σ-smooth if ‖T(fn)Y_‖→0 whenever a sequence of scalar functions (‖fn(⋅)X_‖) is order convergent to 0 in B(Σ). It is shown that a bounded linear operator is σ-smooth if and only if its representing measure is variationally semi-regular, i.e., as An↓∅ (here stands for the semivariation of m on A∈Σ). As an application, we show that the space Lσs(B(Σ,X),Y) of all σ-smooth operators from B(Σ,X) to Y provided with the strong operator topology is sequentially complete. We derive a Banach-Steinhaus type theorem for σ-smooth operators from B(Σ,X) to Y. Moreover, we characterize countable additivity of measures in terms of continuity of the corresponding operators .     

Strong and A-statistical comparisons for sequences

Let T and A be two nonnegative regular summability matrices and W(T,p)∩l^∞ and c_A(b) denote the spaces of all bounded strongly T-summable sequences with index p>0, and bounded summability domain of A, respectively. In this paper we show, among other things, that is a multiplier from W(T,p)∩l^∞ into c_A(b) if and only if any subset K of positive integers that has T-density zero implies that K has A-density zero. These results are used to characterize the A-statistical comparisons for both bounded as well as arbitrary sequences. Using the concept of A-statistical Tauberian rate, we also show that is not a multiplier from W(T,p)∩l^∞ into c_A(b) that leads to a Steinhaus type result.     

On Neumann Operators

We give a sufficient condition for the validity of the implication lim_n→∞ Tⁿx=0⇒∑^∞_n=0Tⁿxconverges, whereT:X→Xis a bounded linear operator,Xis a Banach space, andx∈X. From this condition we derive the results given by other authors. Moreover, we give some properties about the operators that verifies the above implication, which are called Neumann operators.     

Hyperreflexivity and operator ideals

Suppose (B,β) is an operator ideal, and A is a linear space of operators between Banach spaces X and Y. Modifying the classical notion of hyperreflexivity, we say that A is called B-hyperreflexive if there exists a constant C such that, for any T∈B(X,Y) with α=supβ(qTi)<∞ (the supremum runs over all isometric embeddings i into X, and all quotient maps of Y, satisfying qAi=0), there exists a∈A, for which β(T−a)?Cα. In this paper, we give examples of B-hyperreflexive spaces, as well as of spaces failing this property. In the last section, we apply SE-hyperreflexivity of operator algebras (SE is a regular symmetrically normed operator ideal) to constructing operator spaces with prescribed families of completely bounded maps.     

Minimal ranks of some quaternion matrix expressions with applications

Suppose that p(X, Y) = A − BX − X^(∗)B^(∗) − CYC^(∗) and q(X, Y) = A − BX + X^(∗)B^(∗) − CYC^(∗) are quaternion matrix expressions, where A is persymmetric or perskew-symmetric. We in this paper derive the minimal rank formula of p(X, Y) with respect to pair of matrices X and Y = Y^(∗), and the minimal rank formula of q(X, Y) with respect to pair of matrices X and Y = −Y^(∗). As applications, we establish some necessary and sufficient conditions for the existence of the general (persymmetric or perskew-symmetric) solutions to some well-known linear quaternion matrix equations. The expressions are also given for the corresponding general solutions of the matrix equations when the solvability conditions are satisfied. At the same time, some useful consequences are also developed.     

Bifurcation and asymptotic bifurcation for equations involving A-proper mappings with applications to differential equations

Let X, Y be real Banach spaces, T: X → YA-proper, and C: X → Y compact. Section 1 of this paper is devoted to the study of bifurcation and asymptotic bifurcation problems for Eq. $\text{(1): Tx ? λCx = 0}$. In Theorem 1 it is shown that if T(0) = C(0) = 0 and T and C have F-derivatives T₀ and C₀ at 0 with T₀A-proper and injective, then each eigenvalue of T₀x ? λC₀x = 0 of odd multiplicity is a bifurcation point for Eq. (1). Theorem 2 shows that if T and C have asymptotic derivatives T_∞ and C_∞, then each eigenvalue of T_∞x ? λC_∞x = 0 of odd multiplicity is an asymptotic bifurcation point for Eq. (1). Special cases are treated when Y = X and T = I ? F with Fk-ball-contractive or when Y ≠ X and T is either of type (S) or of strongly accretive type. Section 2 is devoted to applications of Theorems 1 and 2 to bifurcation problems involving elliptic operators. The usefulness of Theorems 1 and 2 stems from the fact that they are directly applicable to differential eigenvalue problems without the preliminary reduction of Eq. (1) to equivalent problems involving compact operators. Moreover, in some cases they are applicable in situations to which the known bifurcation results are not applicable.     

Khinchin’s inequality,Dunford-Pettis and compact operators on the space <Emphasis Type="Italic">C</Emphasis>([0, 1], <Emphasis Type="Italic">X</Emphasis>)

We prove that if X, Y are Banach spaces, Ω a compact Hausdorff space and U:C(Ω, X) → Y is a bounded linear operator, and if U is a Dunford-Pettis operator the range of the representing measure G(Σ) ? DP(X, Y) is an uniformly Dunford-Pettis family of operators and ∥G∥ is continuous at Ø. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space C([0, 1], X) with values in c ₀ or l _p, (1 ≤ p < ∞) be Dunford-Pettis and/or compact operators, in which, Khinchin’s inequality plays an important role.     

On additive maps preserving certain semi-Fredholm subsets

Let X and Y be two infinite dimensional real or complex Banach spaces, and let φ: ?(X)?→??(Y) be an additive surjective mapping that preserves semi-Fredholm operators in both directions. In the complex Hilbert space context, Mbekhta and ?emrl [M. Mbekhta and P. ?emrl, Linear maps preserving semi-Fredholm operators and generalized invertibility, Linear Multilinear Algebra 57 (2009), pp. 55–64] determined the structure of the induced map on the Calkin algebra. In this article, we show the following: given an integer n?≥?1, if φ preserves in both directions ? _n (X) (resp., 𝒬 _n (X)), the set of semi-Fredholm operators on X of non-positive (resp., non-negative) index, having dimension of the kernel (resp., codimension of the range) less than n, then φ(T)?=?UTV for all T or φ(T)?=?UT*V for all T, where U and V are two bijective bounded linear, or conjugate linear, mappings between suitable spaces.     

On the m order difference sequence space of generalized weighted mean and compact operators

In this article, using generalized weighted mean and difference matrix of order m, we introduce the paranormed sequence space ?(u, v, p; Δ^(m)), which consist of the sequences whose generalized weighted Δ^(m)-difference means are in the linear space ?(p) defined by I.J. Maddox. Also, we determine the basis of this space and compute its α-, β- and γ-duals. Further, we give the characterization of the classes of matrix mappings from ?(u, v, p, Δ^(m)) to ?_∞, c and c₀. Finally, we apply the Hausdorff measure of noncompacness to characterize some classes of compact operators given by matrices on the space ?_p(u, v, Δ^(m))(1 ≤ p < ∞).     

On spectral properties of a new operator over sequence spaces c and c0

In this work, we classify and calculate spectra such as point spectrum, continuous spectrum and residual spectrum over sequences spaces ?_∞, c${?}_{\infty },\hspace{0.17em}c$ and c₀ according to a new matrix operator W which is obtained by matrix product.     

On complemented copies of c 0 and ℓ ∞

We furnish examples of pairs of Banach spaces X, Y so that none of c ₀ and l _∞ live inside X ^? and Y, but they embed complementably into the space DP(X,Y) of the Dunford–Pettis operators from X into Y.     

Asymptotic centers and best compact approximation of operators intoC(K)

The existence of best compact approximations for all bounded linear operators fromX intoC(K) is related to the behavior of asymptotic centers inX ^*. IfK is just one convergent sequence, the condition is that everyω ^*-convergent sequence inX ^* will have an asymptotic center. We first study this property, solving some open problems in the theory of asymptotic centers. IfK is more “complex,” the asymptotic centers should behave “continuously.” We use this observation to construct operators fromC[0,1] intoC(ω ²) and from ?₁ intoL ₁ without best compact approximation. We also construct spacesX ₁,X ₂, isomorphic to a Hilbert space, and operatorsT ₁,∶X ₁→C(ω ²),T ₂∶?₁→X ₂ without best compact approximations.