Linear Operators: Spectral theory |
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Page 1190
... everywhere defined operator then the statements T * QT and T * = T are equivalent and thus a bounded operator is symmetric if and only if it is self adjoint . If T is an everywhere defined sym- metric operator then T * 2T and thus T ...
... everywhere defined operator then the statements T * QT and T * = T are equivalent and thus a bounded operator is symmetric if and only if it is self adjoint . If T is an everywhere defined sym- metric operator then T * 2T and thus T ...
Page 1212
... everywhere on e ,. Consequently ( B „ f ) ( 2 ) = { s f ( 8 ) W2 + 1 ( 8 , 2 ) ; ( ds ) , fe Li ( Sn , v ) , = n n ... everywhere in S , and since US1 = S this equality must hold v - almost everywhere in S. Q.E.D. 10 DEFINITION . Let W ...
... everywhere on e ,. Consequently ( B „ f ) ( 2 ) = { s f ( 8 ) W2 + 1 ( 8 , 2 ) ; ( ds ) , fe Li ( Sn , v ) , = n n ... everywhere in S , and since US1 = S this equality must hold v - almost everywhere in S. Q.E.D. 10 DEFINITION . Let W ...
Page 1233
... everywhere defined , bounded operator of norm not more than ( 2 ) -1 . Consequently , the series [ * ] ∞ Σ ( λ — λ 。) ” R ( λ 。) ” + 1 n = 0 converges if 12-20 | < | ( 26 ) | . Since T1 is closed , we have -- - ∞ ( T1 — λ1 ) Σ ( 2 ...
... everywhere defined , bounded operator of norm not more than ( 2 ) -1 . Consequently , the series [ * ] ∞ Σ ( λ — λ 。) ” R ( λ 。) ” + 1 n = 0 converges if 12-20 | < | ( 26 ) | . Since T1 is closed , we have -- - ∞ ( T1 — λ1 ) Σ ( 2 ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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(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 satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero