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Trace Class, Widths and the Finite Approximation Property in Banach Space BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 79, Number 1, January 1973 TRACE CLASS, WIDTHS AND THE FINITE APPROXIMATION PROPERTY IN BANACH SPACE BY R. A. GOLDSTEIN AND R. SAEKS Communicated by Robert G. Bartle, June 21, 1972 I. Introduction. Although there have been a number of attempts [3] to define operator classes in Banach space whose properties are analogous to the classical trace class operators of Hubert space [1], [2] it is generally agreed that a satisfactory definition has yet to be achieved [3]. The purpose of the present note is to introduce a new approach to the problem wherein operator widths [2], [4] in Banach space replace the eigenvalues of the Hubert space formulation ; the viability of the approach being illustrated by the formulation of a number of sufficient conditions for an operator to have the finite approximation property in terms of its widths. Moreover, unlike the previous approaches [3] the trace class operators defined via operator widths are representation independent and coincide exactly with the classical definitions in Hilbert space. II. Definitions and results. In the sequel X is a Banach space normed by ||-||, B is the unit ball in X and 5£n is the set of n-dimensional subspaces of B. The nth width, dn{A\ of an operator A on X is defined [4] by (1) dn(A) = inf sup inf||^w — v\\. Le&n ueB veL Classically, Kolmogorov [4] defined the nth width of a set to be a measure of the degree to which the set could be approximated by n-dimensional subspaces, the definition of equation (1) being that of Kolmogorov applied to A(B). Some remarks concerning the sequence {dn{A)} are as follows : (a) d0(A)=\\A\\; (b) {dn(A)} is a nonincreasing sequence and dn(A) -> 0 iff A is a compact operator ; (c) (see for example [2]) for X a Hilbert space and A a compact linear 1/2 1/2 operator on X, set sn(A) = K((A*A) ) = the nth eigenvalue of (A*A) (n = 1,2,...) (these are called the s numbers or characteristic numbers of n A). Then dn(A) = sn+1(A) ( = °> 1,2,...). A is an Hilbert-Schmidt or nuclear operator if the sequence {s„(,4)} is an l2 or lx sequence. AMS (MOS) subject classifications (1970). Primary 47B05, 47B10, 35P05, 35P10, 46B99. Key words and phrases. Widths, trace-class, finite approximation property, d-nuclear, compact operator. Copyright © American Mathematical Society 1973 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use57 58 R. A. GOLDSTEIN AND R. SAEKS [January We define the "trace" classes Dp of compact linear operators on a Banach space as p Dp = ) A compact, linear £ dn(A) < oo L. (In the case of a Hubert space they reduce to the classes Cp or Sp defined in [1] or [2].) Define ||.|L:D.-K by 1/p p \\A\L = Idn(A) .0 THEOREM 1. \\A\\p is a norm on Dp. An operator A e Dl will be called d-nuclear. Before proceeding to our main theorem on the approximation of trace class operators, we introduce another sequence, pn(X\ of positive numbers — which characterize the Banach space X rather than the operator A — defined by pn(X) = sup inf \\P(L)\\, Le^n P(L)ePL. where j£?„ denotes the set of all n-dimensional linear subspaces of X, and PL denotes the set of all projections into the subspace L. (i) 1 S pn(X) S n. (ii) If * is a Hubert space, pn(X) = 1 (Vn). (iii) pn(X) = pn(X*) if X is reflective. (iv) Murray [5] has shown that pn(X) grows linearly with n for X = Lp or lp (p 4 2). We say that A has the finite approximation property iff 3 a sequence of operators, Ani with finite-dimensional ranges, such that \\A — An\\ -> 0 as n -> oo. THEOREM 2. Let A be a compact operator on X such that Urn [dn(A)Pn(X)] = 0. n-* oo Then A has the finite approximation property. Since, in Hilbert space pn(X) = 1, and dn(A) -> 0 for A compact, we trivially obtain the classical result COROLLARY 3. Every compact operator on a Hilbert space has the finite approximation property. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1973] THE FINITE APPROXIMATION PROPERTY IN BANACH SPACE 59 Although we are not able to answer the classical conjecture on the finite approximation property for any compact operator on a Banach space we can observe COROLLARY 4. Every A e Dx has the finite approximation property. 1+e This holds because {dn(A)}elu hence dn(A) = 0(l/n ) (s > 0) and 1+B 8 dn(A)pn(X) = 0(n • l/n ) = 0(n~ ) (n -> oo). The standard examples of nuclear operators in Hubert space have natural analogs in Banach space to which the corollary applies. In particular, the injection maps of Sobolev spaces WQ,P into LP is d-nuclear, and integral operators with smooth kernels on Lp into Lp are d-nuclear. PROOF OF THEOREM 2. dJA) = min max |Ux — L\\ = min max max ||x*Ax|| " LeJSPn||x|| = l LeJ?n \\x\\ = \ x*eL±;\x*\ = \ = min max ||/l*x*|L where X* => LJ is the annihilator of L. Le<en y*eV-\\x*\ = l A LEMMA. P{L)* = 1 - P(L ). Let L e ^£n be the minimizing subspace for dn(A) (which exists since A is compact). Then 3 a projection Pn — P{L) such ||PJ = 0(pn(X)). Set We then have Since Umi = ||1 - P„|| g 1 + ||PB||, we have 2p„(X)dn(A) ^ sup|M*Pix*||/||x*|| = |M*Pi|| = ||(1 - Pn)A\l X* Setting An = PnA, \\A ~ An\\ = 0(dn(A)pn(X)) and the theorem is proved. We conjecture that the condition of Theorem 2 is also necessary. ACKNOWLEDGEMENT. The authors wish to express their gratitude to Professor Abraham Goetz whose insights and suggestions help to guide and formulate their work. REFERENCES 1. N. Dunford and J. T. Schwartz, Linear operators. II: Spectral theory. Selfadjoint operators in Hubert space, Interscience, New York, 1963. MR 32 #6181. 2. I. C. Gohberg and M. G. Kreïn, Introduction to the theory of linear nonse If adjoint operators in Hilbert space, "Nauka", Moscow, 1965; English transi., Transi. Math. Mono­ graphs, vol. 18, Amer. Math. Soc, Providence, R.I., 1969. MR 36 #3137. 3. I. M. Gel'fand and N. Ja. Vilenkin, Generalized functions. Vol. 4: Some applications of harmonic analysis, Fizmatgiz, Moscow, 1961 ; English transi., Academic Press, New York, 1964. MR 26 #4173; MR 30 #4152. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 60 R. A. GOLDSTEIN AND R. SAEKS 4. G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York, 1966. MR 35 #4642. 5. F. J. Murray, On complementary manifolds and projections in spaces Lv and L, Trans. Amer. Math. Soc. 41 (1937), 138-152. 6. L. Nirenberg, Functional analysis, Lecture Notes, New York University, 1961. DEPARTMENT OF MATHEMATICS AND THE DEPARTMENT OF ELECTRICAL ENGINEERING, UNIVERSITY OF NOTRE DAME, NOTRE DAME, INDIANA 46556 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use.
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