ON COMPLETE CONVERGENCE FOR ARRAYS OF ROWWISE INDEPENDENT RANDOM ELEMENTS

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.

We extend a complete convergence theorem for row sums from an array of rowwise independent random variables obtained by Sung, Volodin, and Hu [Statist. Probab. Lett. 71, 303–311 (2005)] to an array of rowwise independent random elements taking values in a real separable Rademacher type p Banach space. This extension is not necessarily valid without the Rademacher type p hypothesis. The hypotheses of the main result do not necessitate the convergence in probability to 0 of the row sums. Corollaries of the main result are obtained and illustrative examples are presented.

We are concerned with the algebraic representation of the spectral maximal spaces for certain classes of decomposable operators on Banach spaces. The main emphasis will be on those operators which admit a functional calculus on a suitable algebra of functions such as, for instance, differentiable functions, Lipschitz functions, or functions with absolutely convergent Fourier series. Using appropriate partitions of unity, we shall give a unified approach to various previous results in this area as well as to certain extensions and variants thereof. In the case of spectral operators, we shall obtain the optimal results without any assumption on the underlying Banach space or the Boolean algebra of projections corresponding to the spectral measure.

For a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak*-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property.

In this paper, apply an established transference principle to obtain the boundedness of certain functional calculi for the sequence of semigroup generators. It is proved that if be the sequence generates 0- semigroups on a Hilbert space, then for each the sequence of operators has bounded calculus for the closed ideal of bounded holomorphic functions on right half–plane. The bounded of this calculus grows at most logarithmically as. As a consequence decay at ∞. Then showed that each sequence of semigroup generator has a so-called (strong) m-bounded calculus for all m∈ℕ, and that this property characterizes the sequence of semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called semigroups, the Hilbert space results actually hold in general Banach spaces.

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If $$ {\frak A} $$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of $$ {\frak A} $$ (i) the quotient $$ {\frak A}/L $$ has the approximation property and (ii) the canonical map from $$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp $$ is open, then $$ {\frak A} $$ is finite-dimensional. As an application, we show that, if $${\frak A}$$ is an amenable Banach algebra whose underlying Banach space is an Lp-space with $$ p\in (1,\infty) $$ such that for each maximal left ideal L the quotient $$ {\frak A}/L $$ has the approximation property, then $$ {\frak A} $$ is finite-dimensional.

It is known that some classes of z-accretive operators A generate Lipschitz continuous semigroups of contractions; that is \\S(t + h)x -S(t)x\\ < L(\\x\\)h/t, 0 si t si t + h si T, x e D(A).If the underlying Banach spaces X and X* are uniformly convex and an w-accretive operator B is bounded, we prove, in particular, that the semigroup generated by A + B is Holder continuous.The proof is based on a result on the structure of accretive operators obtained via the Kuratowski-Ryll-Nardzewski Selection Theorem.Also, we consider some applications of these results to the existence of solutions of u' + Au + Bu 3 Cu, u(0) = ,,.[3; 1, 4.2].

We show that the operator-valued Marcinkiewicz and Mikhlin Fourier multiplier theorem are valid if and only if the underlying Banach space is isomorphic to a Hilbert space.

Let $(x_{nk}), k = 1, \cdots, k_n; n = 1, 2, \cdots$ be a double sequence of infinitesimal random variables which are rowwise independent (i.e., $\lim_{n\rightarrow\infty} \max_{1\leqq k \leqq k_n} P (| x_{nk} | > \epsilon) = 0$ for every $\epsilon > 0,$ and for each $n x_{n1}, \cdots, x_{nk_n}$ are independent). Let $S_n = x_{n1} + \cdots + x_{nk_n} - A_n$ where the $A_n$ are constants and let $F_n(x)$ be the distribution function of $S_n.$ Necessary and sufficient conditions for $F_n(x)$ to converge to a distribution function $F(x)$ are known, and in particular we know that $F(x)$ is infinitely divisible. In this paper we shall investigate the system of infinitesimal, rowwise independent random variables $(| x_{nk} | ^r), r \geqq 1.$ In particular we shall be interested in large values of $r$. Specifically, let $S^r_n = | x_{n1} | ^r + \cdots + | x_{n1} | ^r - B_n(r),$ where $B_n(r)$ are suitably chosen constants. Let $F_n^r(x)$ be the distribution function of $S^r_n.$ Necessary and sufficient conditions for $F_n^r(x)$ to converge $(n \rightarrow \infty)$ to a distribution function $F^r(x)$ are given, and also necessary and sufficient conditions for $F^r(x)$ to converge $(r \rightarrow \infty)$ to a distribution function $H(x)$ are given. The form that $H(x)$ must take is obtained and under rather general conditions it is shown that $H(x)$ is a Poisson distribution. In any case it is shown that $H(x)$ is the sum of two independent random variables, one Gaussian and the other Poisson (including their degenerate cases).

Modified block iterative algorithm for Quasi- ϕ-asymptotically nonexpansive mappings and equilibrium problem in banach spaces

A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is used to obtain optimum bounds of the Rosenthal-Burkholder and Chung types on moments of the martingales in 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on infinite-dimensional martingales, most of the results seem to be new even for one-dimensional martingales. Moreover, the bounds on moments of the Rosenthal-Burkholder type seem to be to a certain extent new even for sums of independent real-valued random variables. Analogous inequalities for (one-dimensional) supermartingales are given.

In this paper we present a decoupling inequality that shows that multivariate $U$-statistics can be studied as sums of (conditionally) independent random variables. This result has important implications in several areas of probability and statistics including the study of random graphs and multiple stochastic integration. More precisely, we get the following result: Let $\{X_j\}$ be a sequence of independent random variables on a measurable space $(\mathscr{J}, S)$ and let $\{X^{(j)}_i\}, j = 1,\ldots,k$, be $k$ independent copies of $\{X_i\}$. Let $f_{i_1i_2\ldots i_k}$ be families of functions of $k$ variables taking $(S \times \cdots \times S)$ into a Banach space $(B, \|\cdots\|)$. Then, for all $n \geq k \geq 2, t > 0$, there exist numerical constants $C_k$ depending on $k$ only so that $P\bigg(\big\|\sum_{1\leq i_1\neq i_2\neq\cdots\neq i_k\leq n} f_{i_1\cdots i_k}(X^{(1)}_{i_1}, X^{(1)}_{i_2}, \ldots, X^{(1)}_{i_k})\big\|\geq t\bigg)$ $\leq C_kP\bigg(C_k\big\|\sum_{1\leq i_1\neq i_2\neq\cdots\neq i_k\leq n} f_{i_1\cdots i_k}(X^{(1)}_{i_1}, X^{(2)}_{i_2}, \ldots, X^{(k)}_{i_k})\big\|\geq t\bigg).$ The reverse bound holds if, in addition, the following symmetry condition holds almost surely: $f_{i_1i_2\cdots i_k}(X_{i_1}, X_{i_2},\ldots,X_{i_k}) = f_{i_{\pi(1)}i_{\pi(2)}\cdots i_{\pi(k)}} (X_{i_{\pi(1)}, X_{i_{\pi(2)}}, \ldots,X_{i_{\pi(k)}}),$ for all permutations $\pi$ of $(1,\ldots,k)$.

0. Introduction. Many features of the theory of commutative W*-algebras in Hilbert space have been shown to extend to appropriate operator algebras in Banach spaces (cf. [2; 3; 7]). Recently an important gap in the theory was closed by Dieudonne [4] who obtained a multiplicity theory in the case that the adjoint X* of the underlying Banach space X (and hence also X) was separable. In this paper a new approach is developed which does not require assumptions of separability and allows us to construct a multiplicity theory also for the algebra of adjoints in X* where the relevant topology is the weak star or %-topology rather than the norm topology. Of the various treatments of multiplicity theory in Hilbert space our development is perhaps closest to that of Kelley [10]. The analogue of a commutative W*-algebra in a Banach space X is the operator algebra a generated in the uniform topology by a complete Boolean algebra 23 of projections in X. For given xeX, the cyclic subspace determined by x is the manifold 9Y(x) =cdm {Ax ACI }. A multiplicity function for f is a function m on e to cardinal numbers, the value m(E) being roughly the smallest number of cyclic subspaces required to span the range of E. The aim of multiplicity theory in this setting is to decompose the identity I in S into the sum of disjoint projections E of uniform multiplicity, i.e. projections such that m(F) =m(E), whenever 0 5 F<E. In ?2 such a decomposition theorem is proved to hold for any function on an abstract B.A. to cardinals which satisfies an appropriate condition of order continuity. Multiplicity functions are constructed respectively for Q3 in X anid Q3* in X* in ??3 and 6 and the cointiniuity condition is verified. Following a detailed analysis of the range of a projectioni of finite uniiform multiplicity, one of the main results of the paper (Corollary 8.5) is obtainied: A projection E in 23 has finite multiplicity n if anid oinly if its adjoint E* in Q* has multiplicity n. We have been unable to decide whether this result remains true for projections of infinite multiplicity. Finally an application is made to the theory of spectral operators of scalar tvpe dleveloped by Dunford [7; 8]. If the resolution of the identity E(.) of

In this paper, we first introduce a class of nonlinear mappings called generalized nonspreading which contains the class of nonspreading mappings in a Banach space and then prove a fixed point theorem, a nonlinear mean convergence theorem of Baillon’s type and a weak convergence theorem of Mann’s type for such nonlinear mappings in a Banach space. Using these theorems, we obtain some fixed point theorems, nonlinear mean convergence theorems and weak convergence theorems in a Banach space.

For a sequence of independent random elements {V n ,n≥1} in a real separable Banach space X, necessary and, separately, sufficient conditions are provided for the strong law of large numbers ∑ i=1 n (V i −c i )/b n →0 almost certainly to hold where {c n ,n≥1} and {b n >0,n≥1} are suitable sequences of centering elements in X and norming constants, respectively. The necessity result extends a real line result of Martikainen[14] to a Banach space setting. The sufficiency result assumes that X is of Rademacher type p (1≤p≤2) and is new even when X is the real line. It is general enough to include as special cases a strong law of Adler, Rosalsky, and Taylor[2] for sums of independent and identically distributed random elements and a strong law of Heyde[9] for sums of independent (real-valued) random variables. Illustrative examples are provided showing that the results are sharp and an example is presented satisfying the hypotheses of the sufficiency result but not those of Heyde's[9] theorem.