Showing papers on "Nuclear operator published in 1974"

Compact and quasinormal composition operators

Abstract: Let C 0 be a composition operator on L2(A), where A is a c-finite measure on a set X. If X is nonatomic, then Ridge proved that no one-to-one composition operator Co with dense range is compact. This result is generalized in the paper by removing one-to-one and dense range conditions. The quasinormal composition operators are also characterized in terms of commutativity with the multiplication operator induced by the Radon-Nikodym derivative of the measure AX-1 with respect to A.

Hermitian and adjoint abelian operators on certain banach spaces

Abstract: Let X be a complex linear space endowed with a semi-inner product [ , ] An operator A on X will be calledHermitian if [Ax, x] is real for all x 6 X; A is said to beadjoint abelian if [Ax, y] = [x, Ay] for all x and yeX Sinceevery Banach space may be given a semi-inner product (notnecessarily unique) which is compatible with the norm, it ispossible to study such operators on general Banach spacesThis paper characterizes Hermitian and adjoint abelian opera-tors on certain Banach spaces which decompose as a directsum of Hubert spaces In particular, the Hermitian operatorsare shown to have operator matrix representations which arediagonal, with the operators on the diagonal being Hermitianoperators on the appropriate Hubert space The class of spacesstudied includes those Banach spaces with hyperorthogonalSchauder bases

Semigroups of scalar type operators on Banach spaces

Abstract: The main result is that if {T(t): t > 0} is a strongly continuous semigroup of scalar type operators on a weakly complete Banach space X and if the resolutions of the identity for T(t) are uniformly bounded in norm, then the infinitesimal generator is scalar type. Moreover, there exists a countably additive spectral measure K( * ) such that T(t) = f exp (Xt)dK(X), for t > 0. This is a direct generalization of the well-known theorem of Sz.-Nagy about semigroups of normal operators on a Hilbert space. Similar spectral representations are given for representations of locally compact abelian groups and for semigroups of unbounded operators. Connections with the theory of hermitian and normal operators on Banach spaces are established. It is further shown that R is the infinitesimal generator of a semigroup of hermitian operators on a Banach space if and only if iR is the generator of a group of isometries.

Characterization of collectively compact sets of linear operators

Abstract: The basic results in this paper show that each collectively compact set of linear operators can be viewed as an equicontinuous collection followed by a single compact operator. This observation not only gives insight into the character of collectively compact sets of linear operators, but also yields easier proofs of many of the results obtained by earlier workers in the field. I* Factorizations of collectively compact operators* A fairly complete treatment, with applications, of collectively compact sets of linear operators is given in the recent book [1] by Anselone. Collectively compact sets of linear operators on normed linear spaces were originally studied by Anselone and Moore [2] in connection with approximate solutions of integral and operator equations. The general properties of such sets of operators, again in normed linear spaces, were studied by Anselone and Palmer in [3] and [4]. Collectively compact sets of linear operators were studied in the more general setting of linear topological spaces by DePree and Higgins [5]. In the current work new characterizations are given for collectively compact sets of operators on a linear topological space.

Best approximation by compact operators. II

Abstract: Introduction. Let H be an infinite-dimensional complex Hubert space, and let 88(H) (resp. ^(H)) be the algebra of all bounded (resp. compact) linear operators on H. It is well known [4], [6] that %>(H) is proximinal in &(H)9 that is, every Te&(H) has a best approximation from the subspace ^(H). Indeed, it has recently been noted by Fakhoury [2] that there is a continuous selection for the associated metric projector. We denote the metric complement of

Perturbation by trace class operators

Abstract: When are two selfadjoint operators unitarily equivalent modulo the trace class? The version of this theorem in which \"trace class\" is replaced by \"compact\" is settled by the Weyl-von Neumann theorem [1]: If A and B are selfadjoint operators, there exists a unitary operator U such that VA U*—B is compact if and only if A and B have the same essential spectrum. The question of trace class equivalence is more delicate and requires a study of additional invariants. Thus the Kato-Rosenblum theorem states: If A and B are selfadjoint operators for which A—B is trace class, then A and B have unitarily equivalent absolutely continuous parts [1]. Our purpose in the present note is to announce the following answer to the question posed above.

Operator valued gramians and inner products on vector valued function spaces

Abstract: The notation of Gramian matrix is generalized to operators between function spaces and then used to provide an operator-valued inner product between pairs of Hilbert space-valued Banach function spaces. In this context projection operators, Schwarz's inequality, and orthonormal series are discussed.

On ideals of compact operators

Abstract: In [1] Brown, Pearcy and Salinas give an affirmative answer to the following question : Given a compact operator T on a separable Hubert space H, is there an ideal A(T) containing T and different from the ideal K of all compact operators ? Their construction relies on some ideas of Von Neumann-Calkin [2] and is rather complicated. The purpose of this brief note is to show that the existence of such a A(T) follows from elementary properties of the .y-numbers of T Recall that the ^-numbers, (sn(T)), are the eigenvalues of (TT*) 1/2 arranged in decreasing order and counting multiplicities. We list the three properties we need : (1) If S, TEK, then sn+m(S+T)^sn(S)+sm(T). (2) If R, S, TeL, then sn(RST)^\\\\R\\\\sn(S)\\\\T\\\\. Here L denotes the space of all bounded linear operators on H. (3) If Te K, then there are orthonormal sets (fn) and (yn) in H such t h a t r = 2 ^ i ^ m / n ^ n . We also use the following fact concerning real sequences : (4) If (/9n) is a nonnegative sequence of real numbers increasing to oo with l//?nec0\\IJ3>>o ^> then there is a positive sequence (oLn)elx such that 2£U &nPn + co a n ( i (/?n°0 * decreasing. Also, there is a decreasing null sequence (yn) such that 2£U 7n n^w + The construction of A(T). Let a„={TeK:2%-i snCO*< + <*>}lt is well known and easy to prove that K\\\\JP>0 QOv. Then pn=\\jsn(T) increases to oo and l/fine c0\\UP>0 lp. Let (ocw) be as in (4) and let

Invarianz gewisser Operatorenklassen unter nichtkommutativen ergodischen Mitteln

Abstract: Let ℋ be a group of *-automorphisms on the algebra of bounded linear operators on a complex Hilbert space H. Then the strongly closed convex hull of the orbit of any compact operator under ℋ consists of compact operators. The same is true if one replaces “compact” by “nuclear”, “Hilbert-Schmidt” or “positive Fredholm”. We further discuss these results in the framework of the noncommutative mean ergodic theorem of KOVACS and SZ#x00FC;CS and formulate an analogous theorem for the algebra of compact operators on a complex Hilbert space.

$p$-absolutely summing operators and the representation of operators on function spaces

Abstract: We introduce a class of p-absolutely summing operators which we call p-extending. We show that for a logmodular function space A(K), an operator T: A(K) -. X is p-extending if and only if there exists a probability measure A on K such that T extends to an isometry T: AP(K, A) -, X. We use this result to give necessary and sufficient conditions under which a bounded linear operator is isometrically equivalent to multiplication by z on d space LP(K, t) and certain Hardy spaces HP(K, At).

Operators with powers essentially similar to those of their adjoints

Abstract: Let T be an operator on a Hilbert space H. In the present note the following result is obtained: If T is an operator such that for some integers p, q, ST*P= ToS+K, where 0 is not in the essential numerical range of S, and K is compact, then for any complex number A in the essential spectrum of T, A*P=Aq Williams [10, Theorem 2] proved that if an operator T on a Hilbert space is such that T*=S-lTS and 0 0 cl(W(S)) then T is similar to a selfadjoint operator. He further showed with different techniques that under this hypothesis on T, the spectrum of T lies on the real line. Taking a clue from this theorem, U.N. Singh and Kanta Mangla [8] obtained the following result: If T is a nonsingular operator such that T* = S-' T-1S and 0 0 cl( W(S)), then T is similar to a unitary operator. So in particular a(T) lies on the unit circle. The author [7] has recently proved the following result using the technique of Williams [10, Theorem 1] which includes this particular case: If T is an operator such that for some integers p, q, T*P=S-1T2S and 0 0 cl(W(S)) then for A E a(T), A*W=Aq. In the present note our interest is to develop these ideas when the canonical image of T*V in the Calkin algebra is similar to that of Tq for some integers p, q. We shall suppose all operators are defined on an infinite dimensional separable Hilbert space H. For an operator T, we write a(T) for the spectrum, Bdry a(T) for the boundary of a(T), w0oo(T) for the isolated points of a(T) that are eigenvalues of finite multiplicity and cl( W(T)) for the closure of the numerical range. If T is the canonical image of T in the Calkin algebra (the algebra of all operators modulo the ideal of compact operators), then a(T) will be called the essential spectrum of T. By the left essential spectrum of T, a1(T), we mean the collection of all A's such that T-Al fails to be left-regular. The numerical range We(T) of T is called the essential numerical range of T. As shown in [9, Theorem 91, Received by the editors September 18, 1972 and, in revised form, March 23, 1973. AMS (MOS) subject classifications (1970). Primary 47A10, 47B20.