Linear Operators: Spectral theory |
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Page 1454
... real axis which is bounded below ; ( b ) the deficiency indices of T are equal . PROOF . To prove ( a ) , note that if T is bounded below , there exists a constant K such that ( Tx , x ) ≥ −K ( x , x ) , x = D ( T ) . The proof of ...
... real axis which is bounded below ; ( b ) the deficiency indices of T are equal . PROOF . To prove ( a ) , note that if T is bounded below , there exists a constant K such that ( Tx , x ) ≥ −K ( x , x ) , x = D ( T ) . The proof of ...
Page 1597
... real axis ( Hartman [ 16 ] ) . ( 22 ) In the interval ( 0 , a ] suppose that q is negative and non- decreasing , and that lim q ( t ) = ∞o . t - 0 Then the essential spectrum of 7 is void ( 6.27 , Sears [ 1 ] ) . ( 23 ) In the interval ...
... real axis ( Hartman [ 16 ] ) . ( 22 ) In the interval ( 0 , a ] suppose that q is negative and non- decreasing , and that lim q ( t ) = ∞o . t - 0 Then the essential spectrum of 7 is void ( 6.27 , Sears [ 1 ] ) . ( 23 ) In the interval ...
Page 1610
... real axis ( NaĬmark [ 5 ] ) . Other conditions which allow approximate determination of the essential spectrum are the following : ( 14 ) Suppose that has the form given in ( 4 ) and that all coefficients are real and eventually non ...
... real axis ( NaĬmark [ 5 ] ) . Other conditions which allow approximate determination of the essential spectrum are the following : ( 14 ) Suppose that has the form given in ( 4 ) and that all coefficients are real and eventually non ...
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