Linear Operators: Spectral theory |
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Page 1291
... T。( τ ) and T1 ( t ) in L2 ( I ) by the formulas ( a ) D ( To ( t ) ) = ( b ) D ( T1 ( t ) ) H ^ ( I ) , H ( 1 ) , To ( t ) f = tf , T1 ( t ) f = tf , fe D ( To ( T ) ) , je D ( T1 ( T ) ) . Observe that To ( t ) and T1 ( t ) are both ...
... T。( τ ) and T1 ( t ) in L2 ( I ) by the formulas ( a ) D ( To ( t ) ) = ( b ) D ( T1 ( t ) ) H ^ ( I ) , H ( 1 ) , To ( t ) f = tf , T1 ( t ) f = tf , fe D ( To ( T ) ) , je D ( T1 ( T ) ) . Observe that To ( t ) and T1 ( t ) are both ...
Page 1400
... of t . Then the deficiency indices of τ are both equal to an integer k and ( a ) for every self adjoint extension T of To ( t ) , the dimension of the null - space { \ T λf } is at most k ; = = ( b ) there exist self adjoint extensions ...
... of t . Then the deficiency indices of τ are both equal to an integer k and ( a ) for every self adjoint extension T of To ( t ) , the dimension of the null - space { \ T λf } is at most k ; = = ( b ) there exist self adjoint extensions ...
Page 1431
... ( T , ( 7 ) ) . This , however , contradicts the result of step ( d ) . ( f ) D ( T1 ( t ' ) ) ≥ D ( T1 ( t ) ) . Clearly = = ( To ( t ) ) = D ( To ( t ' ) ) , so that by step ( e ) D ( To ( t ) ) D ( To ( T ' ) ) . Let D2 = D ( To ( t ) ) = ...
... ( T , ( 7 ) ) . This , however , contradicts the result of step ( d ) . ( f ) D ( T1 ( t ' ) ) ≥ D ( T1 ( t ) ) . Clearly = = ( To ( t ) ) = D ( To ( t ' ) ) , so that by step ( e ) D ( To ( t ) ) D ( To ( T ' ) ) . Let D2 = D ( To ( t ) ) = ...
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