Linear Operators: Spectral theory |
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Page 1190
... everywhere defined operator then the statements T * 2T and TT 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 * = T ...
... everywhere defined operator then the statements T * 2T and TT 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 * = T ...
Page 1233
... everywhere defined , bounded operator of norm not more than ( 0 ) 1 . Consequently , the series [ * ] ∞ Σ ( λ — λ 。) " R ( λ 。) ” + 1 n = 0 converges if 2-20 | < | ( 20 ) ] . Since T1 is closed , we have ∞ 1 +1 ( T1 − 21 ) Σ ( 2 ...
... everywhere defined , bounded operator of norm not more than ( 0 ) 1 . Consequently , the series [ * ] ∞ Σ ( λ — λ 。) " R ( λ 。) ” + 1 n = 0 converges if 2-20 | < | ( 20 ) ] . Since T1 is closed , we have ∞ 1 +1 ( T1 − 21 ) Σ ( 2 ...
Page 1402
... everywhere in 4. 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 ( W , ( , 2 ) ) = 0 μ - almost everywhere in A ...
... everywhere in 4. 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 ( W , ( , 2 ) ) = 0 μ - almost everywhere in A ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero