Linear Operators: General theory |
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Page 486
... operator topology of B ( X , Y ) . PROOF . Let S be the closed unit sphere in X , let T be compact , and let T - T ... Linear combinations of compact linear operators are compact operators , and any product of a compact linear operator ...
... operator topology of B ( X , Y ) . PROOF . Let S be the closed unit sphere in X , let T be compact , and let T - T ... Linear combinations of compact linear operators are compact operators , and any product of a compact linear operator ...
Page 494
... operator T , defined by ( b ) , is a bounded linear operator on C ( S ) to X whose adjoint T * is given by ( d ) . From IV.10.2 we conclude that T * maps the unit sphere of X * into a conditionally weakly compact set of rca ( S ) , and ...
... operator T , defined by ( b ) , is a bounded linear operator on C ( S ) to X whose adjoint T * is given by ( d ) . From IV.10.2 we conclude that T * maps the unit sphere of X * into a conditionally weakly compact set of rca ( S ) , and ...
Page 581
... operator . = 7 Let T be the map in l „ , 1 ≤ p ≤∞o , defined by T [ §1 , §2 , . . . ] [ 2 , 3 , ... ] . Find σ , ( T ) , o , ( T ) , σ ( T ) . 8 If E is a projection operator , find the resolvent R ( 2 ; E ) ex- plicitly in terms of ...
... operator . = 7 Let T be the map in l „ , 1 ≤ p ≤∞o , defined by T [ §1 , §2 , . . . ] [ 2 , 3 , ... ] . Find σ , ( T ) , o , ( T ) , σ ( T ) . 8 If E is a projection operator , find the resolvent R ( 2 ; E ) ex- plicitly in terms of ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ