IN THIS PAPER WE PROVE SEVERAL FUNCTIONAL ANALYTIC PROPERTIES OF THESE DECOMPOSITIONS, AND SHOW HOW THESE PROPERTIES APPLY TO GABOR AND WAVELET SYSTEMS. A CELEBRATED CLASSICAL RESULTSTATES THAT AN OPERATOR U ON A BANACH SPACE IS INVERTIBLE IF IT ISCLOSE ENOUGH TO THE IDENTITY OPERATOR 1 IN THE SENSE THAT |1−U|<1.HERE WE SHOW THAT U ACTUALLY IS INVERTIBLE UNDER A MUCH WEAKERCONDITION.

IN THIS PAPER, WE WANT TO STUDY THE ELEMENTARY OPERATORS OF THE FORM SXS−1 + S−1 XS. USING SOME INEQUALITIES ON THIS OPERATOR, WE CAN CHARACTERIZE SOME OF THE IMPORTANT SUBSPACES OF B(H).

WE STUDY INTEGRAL OPERATORS ON A LARGE FAMILY OF ANALYTIC FUNCTION SPACES, CALLED F(P, Q, S) SPACES. OUR APPROACH FOR THE STUDY OF INTEGRAL OPERATORS IS TO INVESTIGATE SOME RELATED MULTIPLICATION OPERATORS ON F (P, Q, S) TYPE SPACES. AS A CONSEQUENCE OF THIS APPROACH, WE OBTAIN CERTAIN PROPERTIES OF INTEGRAL OPERATORS ON QS SPACES.

THE PURPOSE OF THIS WORK IS TO GIVE SOME NEW RESULTS CONCERNING SPACE OF OPERATORS IN TERMS OF SOME SUBSETS OF BANACH SPACE X. WE WILL GIVE EQUIVALENT CHARACTERIZATION OF BANACH SPACES X IN WHICH EVERY V*-SUBSET OF X IS RELATIVELY COMPACT. WE ALSO DISCUSS SOME APPLICATIONS OF THESE RESULTS TO THE SUBSPACES OF BOUNDED LINEAR OPERATORS.

LET X AND Y BE BANACH SPACES AND N BE A COLLECTION OF SUBSETS IN X.B (X, Y) WILL DENOTE THE BANACH SPACE OF ALL BOUNDED LINEAR OPERATORS FROM X TO Y.