Linear Operators: Spectral theory |
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Page 1316
... satisfies the boundary conditions B * ( f ) = 0 , i = 1 , ... , k * of Lemma 3. Now g coincides with K ( c , · ) in neighborhoods of both a and b , and in view of the remark following Corollary 2.28 we see that K ( c , · ) satisfies the ...
... satisfies the boundary conditions B * ( f ) = 0 , i = 1 , ... , k * of Lemma 3. Now g coincides with K ( c , · ) in neighborhoods of both a and b , and in view of the remark following Corollary 2.28 we see that K ( c , · ) satisfies the ...
Page 1385
... satisfies the boundary condition ƒ ( 0 ) + kf ' ( 0 ) if and only if 1 - k√ − λ 0 ; i.e. , if and only if k is positive and λ = -1 / k2 . Thus , only in case ( iv ) does T have a non - void point spectrum , which consists of the ...
... satisfies the boundary condition ƒ ( 0 ) + kf ' ( 0 ) if and only if 1 - k√ − λ 0 ; i.e. , if and only if k is positive and λ = -1 / k2 . Thus , only in case ( iv ) does T have a non - void point spectrum , which consists of the ...
Page 1602
... satisfies S ' ' \ f ( s ) \ 2 ds = O ( t * ) for some k > 0. Then the point 2 belongs to the essential spectrum of T ( Wintner [ 17 ] ) . ( 49 ) Suppose that the function q is bounded below , and that for some constant k > 0 every ...
... satisfies S ' ' \ f ( s ) \ 2 ds = O ( t * ) for some k > 0. Then the point 2 belongs to the essential spectrum of T ( Wintner [ 17 ] ) . ( 49 ) Suppose that the function q is bounded below , and that for some constant k > 0 every ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero