Linear Operators: Spectral operators |
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Page 2174
... hence is a scalar type operator with real spectrum . ( This follows from Lemma XV.6.1 which implies that the bounded ... Hence we have 1 2 | ett ( s + T ) | = | ettsettт | ≤etts || et | ≤ M1M2 for all te R. Hence if this boundedness ...
... hence is a scalar type operator with real spectrum . ( This follows from Lemma XV.6.1 which implies that the bounded ... Hence we have 1 2 | ett ( s + T ) | = | ettsettт | ≤etts || et | ≤ M1M2 for all te R. Hence if this boundedness ...
Page 2312
... hence for all y in the closure of the range of T. On the other hand , if y is not in the closure of the range of T ... hence closed by Corollary IV.3.2 . Hence it suffices to show that the second of the direct summands is closed . By VII ...
... hence for all y in the closure of the range of T. On the other hand , if y is not in the closure of the range of T ... hence closed by Corollary IV.3.2 . Hence it suffices to show that the second of the direct summands is closed . By VII ...
Page 2357
... Hence = -V V ( P + N ) ( S – XI ) ̄ ” — P ( S — \ I ) − ' + N ( S — \ I ) − " = P ( TMI ) - ' L + N ( S – XI ) - " = is a bounded operator which is compact if P ( T — XI ) - ' is compact ( cf. VI.5.4 ) . In all cases ( a ) , ( b ) ...
... Hence = -V V ( P + N ) ( S – XI ) ̄ ” — P ( S — \ I ) − ' + N ( S — \ I ) − " = P ( TMI ) - ' L + N ( S – XI ) - " = is a bounded operator which is compact if P ( T — XI ) - ' is compact ( cf. VI.5.4 ) . In all cases ( a ) , ( b ) ...
Contents
SPECTRAL OPERATORS | 1924 |
Spectral Operators | 1925 |
Terminology and Preliminary Notions | 1928 |
Copyright | |
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A₁ adjoint operator algebra of projections Amer analytic arbitrary B-algebra B*-algebra B₁ Banach space Boolean algebra Borel sets boundary conditions bounded Borel function bounded linear operator bounded operator closed operator commuting compact complex numbers complex plane converges Corollary countably additive Definition denote dense differential operator Doklady Akad domain eigenvalues elements equation exists finite number follows from Lemma follows from Theorem formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality integral invariant inverse L₁ Lebesgue Lemma Math multiplicity Nauk SSSR norm operators in Hilbert perturbation polynomial PROOF properties prove quasi-nilpotent resolution Russian S₁ satisfies scalar type operator scalar type spectral Section sequence shows spectral measure spectral operator spectral theory spectrum subset subspace Suppose trace class type spectral operator unbounded uniformly bounded unique vector zero