Abstract: Let X be a Banach space and let T be a bounded linear operator acting on X. Atkinson's well known theorem says that T is a Fredholm operator if and only if its projection in the algebra L(X)=F0(X) is invertible, where F0(X) is the ideal of nite rank operators in the algebra L(X) of bounded linear operators acting on X. In the main result of this paper we establish an Atkinson-type theorem for B-Fredholm operators. More precisely we prove that T is a B-Fredholm operator if and only if its projection in the algebra L(X)=F0(X) is Drazin invertible. We also show that the set of Drazin invertible elements in an algebra A with a unit is a regularity in the sense dened by Kordula and M (8).

Abstract: We prove that the Hausdor operator generated by a function is bounded on the real Hardy space H p (R), 0 0( 1 jF (x + it)j p dx) 1=p 0g, with the norm kfkHp(R) = kFk H p (R 2) . The Fourier transform of a function f(x) in R is given by (f(x)) ^ ( ) = b

Abstract: We present a general spectral decomposition technique for bounded solu- tions to inhomogeneous linear periodic evolution equations of the form _ x = A(t)x+f(t) ( ), with f having precompact range, which is then applied to nd new spectral criteria for the existence of almost periodic solutions with specic spectral properties in the resonant case where e i sp(f) may intersect the spectrum of the monodromy operator P of ( ) (here sp(f) denotes the Carleman spectrum of f). We show that if ( ) has a bounded uniformly continuous mild solution u and (P )nei sp(f) is closed, where (P ) denotes the part of (P ) on the unit circle, then ( ) has a bounded uniformly continuous mild solution w such that ei sp(w) = ei sp(f). Moreover, w is a \spectral component" of u. This allows us to solve the general Massera-type problem for almost periodic solutions. Various spectral criteria for the existence of almost periodic and quasi-periodic mild solutions to ( ) are given.

Abstract: Logarithmic convexity of a measure of weak noncompactness for bounded linear operators under Calderón’s complex interpolation is proved. This is a quantitative version for weakly noncompact operators of the following: if T : A0 → B0 or T : A1 → B1 is weakly compact, then so is T : A[θ] → B[θ] for all 0 < θ < 1, where A[θ] and B[θ] are interpolation spaces with respect to the pairs (A0, A1) and (B0, B1). Some formulae for this measure and relations to other quantities measuring weak noncompactness are established.

Abstract: .. . . . . . . . . . . . . .. . ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .... .... .. . ;

Abstract: The Kaczmarz algorithm of successive projections suggests the following concept. A sequence (ek) of unit vectors in a Hilbert space is said to be effective if for each vector x in the space the sequence (xn) converges to x where (xn) is defined inductively: x0 = 0 and xn = xn−1 + αnen, where αn = 〈x − xn−1, en〉. We prove the effectivity of some sequences in Hilbert spaces. We generalize the concept of effectivity to sequences of vectors in Banach spaces and we prove some results for this more general concept. Introduction. In 1937 S. Kaczmarz [2] proved that given nonzero vectors (ek) in a finite-dimensional vector space and numbers (ck), a solution (if it exists) to the system of linear equations 〈x, ek〉 = ck, k = 1, . . . , N , can be obtained as the limit of the sequence defined inductively by x0 = 0, xn = xn−1 + cn − 〈xn−1, en〉 〈en, en〉 en (here the sequences (en), (cn) are extended to infinite sequences by making them N -periodic). Since then, the method (also known as the “row-action method”) has been extended and modified in many ways. (For a review of the method, its generalizations, outlines of applications and references see [1].) In this paper we consider the following generalization of the method: Let X be a real or complex Banach space. Let (ek, fk) be a sequence such that ek ∈ X, fk ∈ X∗, fk(ek) = 1 for all positive integers k (X∗ denotes the dual space). The Kaczmarz algorithm of approximation of elements of X constructs for each x ∈ X an approximating sequence xn defined by the iterative procedure x0 = 0, xn = xn−1 + fn(x− xn−1)en. 2000 Mathematics Subject Classification: 41A65, 60G25, 60H25. Research of S. Kwapień supported in part by Polish Grant KBN 2 PO3A 043 15.

Abstract: We use a non-commutative generalization of the Banach Principle to show that the classical individual ergodic theorem for subsequences generated by means of uniform sequences can be extended to the von Neumann algebra setting. 0. Introduction. The study of almost everywhere convergence of the ergodic averages in the non-commutative setting was initiated by a number of authors among whom we mention Lance (5) and Yeadon (11). Individual ergodic theorems have been established for algebras with states as well as for algebras equipped with a seminite trace. The study of almost everywhere convergence of weighted and subsequential averages in von Neumann alge- bras is relatively new. So far, not much is known in this direction. Recently, a non-commutative analog of the classical Banach Principle, on convergence of sequences of measurable functions generated by a sequence of linear maps on L p -spaces, was established in (3). It is expected that, as in the commuta- tive case, this principle will be instrumental in obtaining various convergence results for the averages in non-commutative setting. In (8), an individual er- godic theorem for subsequences was proved, where the proof was based on application of the \commutative" Banach Principle. In this paper we use the ergodic theorem of Yeadon (11) together with the results of (3), adjusted to the bilateral almost uniform convergence, to show that the main result of (8) also holds in the vNa setting. 1. Preliminaries. Let M be a von Neumann algebra (vNa) acting on a Hilbert space H. Let I be the unit of M, and let be a faithful normal seminite trace on M. Denote by P (M) the complete lattice of all projec- tions in M. Let A(M) be the set of all closed operators aliated with M. An operator x2 A(M) is said to be -measurable if for each " > 0 there

Abstract: 9 p. : il. ; 30 cm. Documento de trabajo (Universidad de San Andres. Departamento de Matematica y Ciencias) ; 14. "Marzo 2000". Incluye referencias bibliograficas (p. 8-9).

Abstract: Let {β(n)}n=0 be a sequence of positive numbers and 1 ≤ p <∞. We consider the space `(β) of all power series f(z)= ∑∞ n=0 f̂(n)z n such that ∑∞ n=0 |f̂(n)||β(n)| < ∞. We give some sufficient conditions for the multiplication operator, Mz , to be unicellular on the Banach space `(β). This generalizes the main results obtained by Lu Fang [1]. Introduction. First, we generalize some definitions from [4]. Let {β(n)} be a sequence of nonzero complex numbers with β(0) = 1 and 1 ≤ p < ∞. We consider the space of sequences f = {f̂(n)}n=0 such that ‖f‖ = ‖f‖pβ = ∞ ∑ n=0 |f̂(n)||β(n)| <∞. The notation f(z) = ∑∞ n=0 f̂(n)z n will be used whether or not the series converges for any value of z. These are called formal power series. Let `(β) denote the space of such formal power series. For 1 < p < ∞, `(β) ∼= L(μ) where μ is the σ-finite measure defined on the positive integers by μ(K) = ∑ n∈K β(n) , K ⊆ N ∪ {0}. So `(β) is a reflexive Banach space ([3]) and (`p(β))∗ = `(β) where β = {β(n)}n ([6]). Let f̂k(n) = δnk. So fk(z) = z and then {fk}k is a basis such that ‖fk‖ = |β(k)|. Now consider Mz, the operator of multiplication by z on `(β): (Mzf)(z) = ∞ ∑

Abstract: We give a common generalization of the Walsh system, Vilenkin system, the character system of the group of 2-adic (m-adic) integers, the product system of normalized coordinate functions for continuous irreducible unitary representations of the coordinate groups of noncommutative Vilenkin groups, the UDMD product systems (defined by F. Schipp) and some other systems. We prove that for integrable functions σnf → f (n → ∞) a.e., where σnf is the nth (C, 1) mean of f . (For the character system of the group of m-adic integers, this proves a more than 20 years old conjecture of M. H. Taibleson [24, p. 114].) Define the maximal operator σ∗f := supn |σnf |. We prove that σ∗ is of type (p, p) for all 1 < p ≤ ∞ and of weak type (1, 1). Moreover, ‖σ∗f‖1 ≤ c‖f‖H , where H is the Hardy space. Introduction and examples. Denote by N the set of natural numbers and by P the set of positive integers. Let m := (mk : k ∈ N) be a sequence of positive integers such that mk ≥ 2 for k ∈ N, and let Gmk be a set of cardinality mk. Suppose that each (coordinate) set has the discrete topology and the measure μk which maps every singleton of Gmk to 1/mk (μk(Gmk) = 1) for k ∈ N. Let Gm be the compact set formed by the complete direct product of Gmk equipped with the product topology and product measure (μ). Thus each x ∈ Gm is a sequence x := (x0, x1, . . .), where xk ∈ Gmk , k ∈ N. Then Gm is called a Vilenkin space. It is a compact totally disconnected space, with normalized regular Borel measure μ. The Vilenkin space Gm is said to be bounded if the generating system m is bounded. Throughout this paper we assume the boundedness of Gm; moreover, c, cp denote absolute constants, the latter can depend (only) on p. 2000 Mathematics Subject Classification: Primary 42C10; Secondary 42C15, 43A75, 40G05. Research supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. F020334, by the Hungarian “Művelődési és Közoktatási Minisztérium”, grant no. FKFP 0710/1997, 0182/2000, and by the Bolyai Fellowship of the Hungarian Academy of Sciences, grant no. BO/00320/99. [101] 102 G. Gát A neighborhood base of Gm can be given as follows: I0(x) := Gm, In(x) := {y = (yi, i ∈ N) ∈ Gm : yi = xi for i < n} for x ∈ Gm, n ∈ P. Then I := {In(x) : n ∈ N, x ∈ Gm} is the set of intervals on Gm. Denote by L(Gm) the usual Lebesgue spaces (with norms ‖ · ‖p) (1 ≤ p ≤ ∞), by An the σ-algebra generated by the sets In(x) (x ∈ Gm) and by En the conditional expectation operator with respect to An (n ∈ N). The maximal Hardy space H(Gm) is defined by means of the maximal function f∗ := supn |Enf | (f ∈ L(Gm)): f is said to be in H(Gm) if f∗ ∈ L(Gm). Then H(Gm) is a Banach space with the norm ‖f‖H1 := ‖f‖1. This definition is suitable if the sequence m is bounded. In this case H(Gm) has an atomic structure (for the dyadic case (mk = 2, k ∈ N) see [21, p. 104] and for the general case see [22, p. 92]). A function g ∈ L∞(Gm) is an atom if either g = 1 or supp g ⊂ In(x), In(x) g dμ = 0, and ‖g‖∞ ≤ 1/μ(In(x)) for some x ∈ Gm, n ∈ N. By definition, f ∈ H(Gm) iff f = ∑∞ i=0 λigi, where ∑∞ i=0 |λi| <∞, λi ∈ C and gi is an atom (i ∈ N). Then H(Gm) is a Banach space with the norm ‖f‖H := inf ∞ ∑ i=0 |λi|, where the infimum is taken over all decompositions f = ∑∞ i=0 λigi as above. If the sequence m is bounded (in this paper this is supposed), then H(Gm) = H(Gm), moreover, the two norms are equivalent. (If the sequence m is not bounded, then the situation changes [22].) We say that an operator T : L1 → L0 (where L(Gm) is the space of measurable functions on the Vilenkin space Gm) is of type (p, p) (for 1 ≤ p ≤ ∞) if ‖Tf‖p ≤ cp‖f‖p for all f ∈ L(Gm) and the constant cp depends only on p; T is of type (H,L) if ‖Tf‖1 ≤ c‖f‖H for all f ∈ H(Gm); and T is of weak type (1, 1) if μ(|Tf | > λ) ≤ c‖f‖1/λ for all f ∈ L(Gm) and λ > 0. Let M0 := 1 and Mk+1 := mkMk for k ∈ N be the so-called generalized powers. Then every n ∈ N can be uniquely expressed as n = ∞k=0 nkMk, 0 ≤ nk < mk, nk ∈ N. The sequence (n0, n1, . . .) is called the expansion of n with respect to m. We often use the following notations. Let |n| := max{k ∈ N : nk 6= 0} (that is, M|n| ≤ n < M|n|+1) and n(k) = ∑∞ j=k njMj . Next we introduce on Gm an orthonormal system which we call a Vilenkin-like system. (C, 1) summability for Vilenkin-like systems 103 Complex-valued functions r k : Gm → C which we call generalized Rademacher functions have the following properties: (i) r k is Ak+1-measurable (i.e. r k (x) depends only on x0, . . . , xk (x ∈ Gm)), r0 k = 1 for all k, n ∈ N. (ii) If Mk is a divisor of n and l and if n(k+1) = l(k+1) (k, l, n ∈ N), then Ek(r k r l k) = { 1 if nk = lk, 0 if nk 6= lk (z is the complex conjugate of z). (iii) If Mk+1 is a divisor of n (that is, n = nk+1Mk+1 +nk+2Mk+2 + . . .+ n|n|M|n|), then mk−1 ∑ j=0 |rk k (x)| = mk for all x ∈ Gm. (iv) There exists a δ > 1 for which ‖rn k‖∞ ≤ √ mk/δ. Define a Vilenkin-like system ψ = (ψn : n ∈ N) as follows: ψn := ∞ ∏ k=0 r (k) k , n ∈ N. (Since r0 k = 1, we have ψn = ∏|n| k=0 r n k .) Example A (the Vilenkin and Walsh systems). Let Gmk := Zmk be the mkth (2 ≤ mk ∈ N) discrete cyclic group (k ∈ N). That is, Zmk can be represented by the set {0, 1, . . . ,mk− 1}, where the group operation is mod mk addition and every subset is open. The group operation (+) on Gm is coordinatewise addition. Gm is called a Vilenkin group. The Vilenkin group for which mk = 2 for all k ∈ N is the Walsh–Paley group. In this case let r k (x) := (exp(2πıxk/mk)) nk , where ı := √ −1, x ∈ Gm. The system ψ := (ψn : n ∈ N) is the Vilenkin system, where ψn := ∏∞ k=0 r n k = ∏∞ k=0 r nkMk k . For the Vilenkin group with mk = 2 for all k ∈ N, we get the Walsh–Paley system. Since |rn k | = 1, (iii) and (iv) are trivial and so are (i) and (ii). For more on the Vilenkin and Walsh systems and groups see e.g. [21, 1]. Example B (the group of 2-adic (m-adic) integers). Let Gmk := {0, 1, . . . ,mk − 1} for all k ∈ N. Define on Gm the following (commutative) addition: Let x, y ∈ Gm. Then x+y = z ∈ Gm is defined in a recursive way. First, x0 +y0 = t0m0 +z0, where (of course) z0 ∈ {0, 1, . . . ,m0−1} and t0 ∈ N. Suppose that z0, . . . , zk and t0, . . . , tk have been defined. Then write xk+1 + yk+1 + tk = tk+1mk+1 + zk+1, where zk+1 ∈ {0, 1, . . . ,mk+1−1} and tk+1 ∈ N. Then Gm is called the group of m-adic integers (if mk = 2 for all 104 G. Gát k ∈ N, then these are the 2-adic integers). In this case let r k (x) := ( exp ( 2πı ( xk mk + xk−1 mkmk−1 + . . .+ x0 mkmk−1 . . .m0 )))nk . Let ψn := ∏∞ k=0 r n k = ∏∞ k=0 r nkMk k . Then the system ψ := (ψn : n ∈ N) is the character system of the group of m-adic (2-adic if mk = 2 for each k ∈ N) integers. Since |rn k | = 1, (i), (iii) and (iv) are trivial. (ii) is also easy to see and well known [22, p. 91]. For more on the group of m-adic integers see e.g. [9, 15, 24]. Example C (noncommutative Vilenkin groups). Let σ be an equivalence class of continuous irreducible unitary representations of a compact group G. Denote by Σ the set of all such σ. Then Σ is called the dual object of G. The dimension of a representation U (σ), σ ∈ Σ, is denoted by dσ and we let u (σ) i,j (x) := 〈U (σ) x ξi, ξj〉, i, j ∈ {1, . . . , dσ}, be the coordinate functions for U (σ), where ξ1, . . . , ξdσ is an orthonormal basis in the representation space of U (σ). (For the notations see [9, Vol. 2, p. 3].) According to the Weyl–Peter theorem (see e.g. [9, Vol. 2, p. 24]), the system of functions √ dσu (σ) i,j , σ ∈ Σ, i, j ∈ {1, . . . , dσ}, is an orthonormal basis for L2(G). If G is a finite group, then Σ is also finite. If Σ := {σ1, . . . , σs}, then |G| = dσ1 + . . .+ dσs . Let Gmk be a finite group of order mk, k ∈ N. Let {rk k : 0 ≤ s < mk} be the set of all normalized coordinate functions of the group Gmk and suppose that r0 k ≡ 1. Thus for every 0 ≤ s < mk there exists a σ ∈ Σk and i, j ∈ {1, . . . , dσ} such that rk k = √ dσu (σ) i,j (x) (x ∈ Gmk). Set r k := r nkMk k . Let ψ be the product system of r j k, namely ψn(x) := ∞ ∏ k=0 r (k) k (xk) (x ∈ Gm), where n is of the form n = ∑∞ k=0 nkMk and x = (x0, x1, . . .). We remark that if Gmk is the discrete cyclic group of order mk, k ∈ N, then Gm coincides with the Vilenkin group, and ψ is the Vilenkin system with respect to the corresponding order [8, 21, 26, 1]. In [8] it is proved that the system ψ has the properties (i)–(iii). Moreover, (iv) is satisfied because mk = |Gmk | = dσk,1 + . . . + d 2 σk,sk , where {σk,i : i = 1, . . . , ks} = Σk (the dual object of Gmk) and dσk,i is the dimension of σk,i. We have ‖r k‖∞ ≤ √ d, where d is one of dσk,i and since d is a divisor of mk [9, Vol. 2, p. 44], [8] and at least one of dσk,i is 1, it follows that d < √ mk. Since m is bounded, we conclude (C, 1) summability for Vilenkin-like systems 105 that there exists a δ > 1 (possibly depending on the sequence m) such that (iv) holds for all n, k ∈ N. For more on this system and noncommutative Vilenkin groups see [8, 6]. Example D (a system in number theory). Let r k (x) := exp ( 2πı ∞ ∑

Abstract: In this note we construct Swiss cheeses X such that R(X) is non-regular but such that R(X) has no non-trivial Jensen measures. We also construct a non-regular uniform algebra with compact, metrizable character space such that every point of the character space is a peak point.

Abstract: We prove two-weight norm inequalities in R n for the minimal operator mf(x) = inf Q3x 1 jQj Q jfjdy;

Abstract: We introduce the notion of a local n-times integrated C-semigroup, which unifies the classes of local C-semigroups, local integrated semigroups and local C-cosine functions. We then study its relations to the C-wellposedness of the (n+1)-times integrated Cauchy problem and second order abstract Cauchy problem. Finally, a generation theorem for local n-times integrated C-semigroups is given.

Abstract: Parabolic wavelet transforms associated with the singular heat operators +@=@t andI +@=@t, where = Pn=1 @ 2 =@x 2 +(2 =xn)@=@xn, are introduced. These transforms are dened in terms of the relevant generalized translation operator. An analogue of the Calder on reproducing formula is established. New inversion formulas are obtained for generalized parabolic potentials representing negative powers of the singular heat operators.

Abstract: “Weyl’s theorem” for an operator on a Hilbert space is a statement that the complement in the spectrum of the “Weyl spectrum” coincides with the isolated eigenvalues of finite multiplicity. In this paper we consider how Weyl’s theorem survives for polynomials of operators and under quasinilpotent or compact perturbations. First, we show that if T is reduced by each of its finite-dimensional eigenspaces then the weyl spectrum obeys the spectral mapping theorem, and further if T is reduction-isoloid then for every polynomial p, Weyl’s theorem holds for p(T ). The results on perturbations are as follows. If T is a “finite-isoloid” operator and if K commutes with T and is either compact or quasinilpotent then Weyl’s theorem is transmitted from T to T +K. As a non-commutative perturbation theorem, we also show that if the spectrum of T has no holes and at most finitely many isolated points, and if K is a compact operator then Weyl’s theorem holds for T +K when it holds for T . Introduction. H. Weyl [22] examined the spectra of all compact perturbations T +K of a hermitian operator T and discovered that λ ∈ σ(T + K) for every compact operator K if and only if λ is not an isolated eigenvalue of finite multiplicity in σ(T ). Today this result is known as Weyl’s theorem, and it has been extended from hermitian operators to hyponormal operators and to Toeplitz operators by L. Coburn [7], to several classes of operators including seminormal operators by S. Berberian [2],[3], and to a few classes of Banach space operators [15],[17]. Weyl’s theorem may fail for even the square of T when it holds for T (see [18, Example 1]). In [14], it was shown that Weyl’s theorem holds for polynomials of hyponormal operators. The first aim of this paper is to extend this result via “Berberian” spectra. On the other hand, Weyl’s theorem is liable to fail under “small” perturbations if “small” is interpreted in the sense of compact or quasinilpotent. Recently Weyl’s theorem under small perturbations has been considered in [11],[12],[13], and [18]. The second aim of this paper is to explore how Weyl’s theorem survives under quasinilpotent or compact perturbations. Throughout this paper let H denote an infinite dimensional separable Hilbert space. Let L(H) denote the algebra of bounded linear operators on H and let K(H) denote the closed ideal of compact operators on H. If T ∈ L(H) write ρ(T ) for the resolvent set of T ; σ(T ) for the spectrum of T ; π0(T ) for the set of eigenvalues of T ; π0f (T ) for the eigenvalues of finite multiplicity; π0i(T ) for the eigenvalues of infinite multiplicity. An operator T ∈ L(H) is said to be Fredholm if T−1(0) and T (H)⊥ are both finite-dimensional. The index of a Fredholm operator T ∈ L(H), denoted ind (T ), is given by ind (T ) = dimT−1(0)− dimT (H)⊥ (= dimT−1(0)− dimT ∗−1(0)). 2000 Mathematics Subject Classification. Primary 47A10,47A53,47A55

Abstract: A metric space (M;d) is said to have the small ball property (sbp) if for every "0 > 0 it is possible to write M as the union of a sequence (B(xn;rn)) of closed balls such that the rn are smaller than "0 and limrn = 0. We study permanence properties and examples of sbp. The main results of this paper are the following: 1. Bounded convex closed sets in Banach spaces have sbp only if they are compact. 2. Precisely the nite- dimensional Banach spaces have sbp. (More generally: a complete metric group has sbp i it is separable and locally compact.) 3. Let B be a boundary in the bidual of an innite-dimension al Banach space. Then B does not have sbp. In particular the set of extreme points in the unit ball of an innite-dimension al reexiv e Banach space fails to have sbp.

Abstract: Let U be a trigonometrically well-bounded operator on a Banach space X, and denote byfAn(U)g 1=1 the sequence of (C; 2) weighted discrete ergodic averages of U, that is, An(U) = 1 n X 0