Linear Operators: General theory |
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Page 566
... resolvent function of T , or simply the resolvent of T. LEMMA . The resolvent set o ( T ) is open . The function R ( 2 ; T ) is analytic in o ( T ) . PROOF . Let be a fixed point in o ( T ) , and let μ be any complex number with μ ...
... resolvent function of T , or simply the resolvent of T. LEMMA . The resolvent set o ( T ) is open . The function R ( 2 ; T ) is analytic in o ( T ) . PROOF . Let be a fixed point in o ( T ) , and let μ be any complex number with μ ...
Page 591
... resolvent set of S + N and 00 R ( 2 ; S + N ) = R ( 2 ; S ) + 1Nn . n = 0 Thus the function f of Theorem 10 is analytic on a neighborhood of o ( S + N ) as stated . Now let C denote the union of a finite collection C1 , . . . , Cn of ...
... resolvent set of S + N and 00 R ( 2 ; S + N ) = R ( 2 ; S ) + 1Nn . n = 0 Thus the function f of Theorem 10 is analytic on a neighborhood of o ( S + N ) as stated . Now let C denote the union of a finite collection C1 , . . . , Cn of ...
Page 607
... resolvent ( 21 - T ) -1 in the neighborhood of a pole . A special case of Theorem 1.9 is due to Weyr [ 1 ] , and in full generality it was proved by Hensel [ 1 ] . In regard to Theorem 1.10 , Frobenius [ 3 ; p . 11 ] stated that if f is ...
... resolvent ( 21 - T ) -1 in the neighborhood of a pole . A special case of Theorem 1.9 is due to Weyr [ 1 ] , and in full generality it was proved by Hensel [ 1 ] . In regard to Theorem 1.10 , Frobenius [ 3 ; p . 11 ] stated that if f is ...
Contents
Preliminary Concepts | 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 ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ