Linear Operators, Part 2 |
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Page 1190
... everywhere defined operator then the statements T * 2T 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 * T and thus T * T. By ...
... everywhere defined operator then the statements T * 2T 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 * T and thus T * T. By ...
Page 1212
... everywhere on e . Consequently ( B „ f ) ( 2 ) = [ § ̧ f ( 8 ) W2 + 1 ( 8 , 2 ) y ( ds ) , ƒɛ L1 ( Sn , v ) , - λεεη ... everywhere in S , and since US , S this equality must hold v - almost everywhere in S. Q.E.D. 10 DEFINITION . Let W ...
... everywhere on e . Consequently ( B „ f ) ( 2 ) = [ § ̧ f ( 8 ) W2 + 1 ( 8 , 2 ) y ( ds ) , ƒɛ L1 ( Sn , v ) , - λεεη ... everywhere in S , and since US , 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 ( -1 . Consequently , the series [ * ] ∞ Σ ( λ — λ 。) ” R ( 2 。) ” + 1 n = 0 converges if 2-20 | < | J ( 26 ) | . Since T , is closed , we have -- ∞ 1 ( T1 — λ1 ) Σ ( 2—20 ) ...
... everywhere defined , bounded operator of norm not more than ( -1 . Consequently , the series [ * ] ∞ Σ ( λ — λ 。) ” R ( 2 。) ” + 1 n = 0 converges if 2-20 | < | J ( 26 ) | . Since T , is closed , we have -- ∞ 1 ( T1 — λ1 ) Σ ( 2—20 ) ...
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 Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm 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 transform unique unitary vanishes vector zero