Linear Operators: Spectral theory |
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Page 1144
... resolvent of T satisfies the inequality | R ( 2 ; T ) | = O ( 2 - N ) as 2.0 along any of the arcs y . Then the subspace sp ( T ) contains the subspace TNS . Similarly , by arguing as in the proofs of Corollary 6.30 and Corollary 6.31 ...
... resolvent of T satisfies the inequality | R ( 2 ; T ) | = O ( 2 - N ) as 2.0 along any of the arcs y . Then the subspace sp ( T ) contains the subspace TNS . Similarly , by arguing as in the proofs of Corollary 6.30 and Corollary 6.31 ...
Page 1187
... resolvent set p ( T ) of an operator T is defined to be the set of all complex numbers such that ( 1 - T ) -1 exists as an everywhere defined bounded operator . For λ in p ( T ) the symbol R ( 2 ; T ) will be used for the resolvent ...
... resolvent set p ( T ) of an operator T is defined to be the set of all complex numbers such that ( 1 - T ) -1 exists as an everywhere defined bounded operator . For λ in p ( T ) the symbol R ( 2 ; T ) will be used for the resolvent ...
Page 1422
... resolvent set of S. Since , by Lemma XII.1.3 , the resolvent set is open , μ is not in the resolvent set . Suppose μo < ∞ , and let μn be a sequence of real numbers approaching from below . By [ †† ] , R ( —μ „ i ; S ) | ≤μ‚1 . It ...
... resolvent set of S. Since , by Lemma XII.1.3 , the resolvent set is open , μ is not in the resolvent set . Suppose μo < ∞ , and let μn be a sequence of real numbers approaching from below . By [ †† ] , R ( —μ „ i ; S ) | ≤μ‚1 . It ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero