Linear Operators, Part 2 |
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Page 1027
... belongs to the spectrum of both T and ET . Suppose that 10 belongs to the spectrum of T. Since T is compact , Theorem VII.4.5 shows that is an eigenvalue and hence for some non - zero x in H we have Tx and hence , since T TE , we have ...
... belongs to the spectrum of both T and ET . Suppose that 10 belongs to the spectrum of T. Since T is compact , Theorem VII.4.5 shows that is an eigenvalue and hence for some non - zero x in H we have Tx and hence , since T TE , we have ...
Page 1116
... belongs to the Hilbert - Schmidt class C2 . If we let Aq ; = y / 2p , then A is plainly self adjoint and A belongs i i to the class C ,, where r ( 1 - p / 2 ) = p , i.e. , r = p ( 1 - p / 2 ) -1 . Thus , by Lemma 9 , TBA belongs to the ...
... belongs to the Hilbert - Schmidt class C2 . If we let Aq ; = y / 2p , then A is plainly self adjoint and A belongs i i to the class C ,, where r ( 1 - p / 2 ) = p , i.e. , r = p ( 1 - p / 2 ) -1 . Thus , by Lemma 9 , TBA belongs to the ...
Page 1134
... belongs to the range of the projection E. Thus ( n ) . ( T " ƒ , f ) = c ” ( b — a ) . Since T is quasi - nilpotent ... belong to the same interval of the complement of C. 11 Conversely , we shall suppose that K11 ( s , t ) = 0 if s and ...
... belongs to the range of the projection E. Thus ( n ) . ( T " ƒ , f ) = c ” ( b — a ) . Since T is quasi - nilpotent ... belong to the same interval of the complement of C. 11 Conversely , we shall suppose that K11 ( s , t ) = 0 if s and ...
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 domain 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 linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma 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