Linear Operators: Spectral theory |
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Page 1190
... everywhere defined operator then the statements T * 2 T 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 * 2 T 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 1233
... everywhere defined , bounded operator of norm not more than ( 2 ) . Consequently , the series [ * ] ∞ Σ ( 2—20 ) " R ( 2 ) ” + 1 n = 0 converges if 2-2 。| < | ( 26 ) ] . Since T1 is closed , we have ∞ ( T1 — λ1 ) Σ ( λ — λ 。) ” R ...
... everywhere defined , bounded operator of norm not more than ( 2 ) . Consequently , the series [ * ] ∞ Σ ( 2—20 ) " R ( 2 ) ” + 1 n = 0 converges if 2-2 。| < | ( 26 ) ] . Since T1 is closed , we have ∞ ( T1 — λ1 ) Σ ( λ — λ 。) ” R ...
Page 1402
... everywhere in A. The proof of Theorem 5.4 will then apply with evident slight modifications to show that if B ( f ) = 0 is a boundary condition satisfied by all ƒ e D ( T ) , we have B ( W1 ( • , 2 ) ) = 0 μ - almost everywhere in A ...
... everywhere in A. The proof of Theorem 5.4 will then apply with evident slight modifications to show that if B ( f ) = 0 is a boundary condition satisfied by all ƒ e D ( T ) , we have B ( W1 ( • , 2 ) ) = 0 μ - almost everywhere in A ...
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