Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 1027
... belongs to the spectrum of both T and ET . Suppose that λ 0 belongs to the spectrum of T. Since T is compact , Theorem VII.4.5 shows that 2 is an eigenvalue and hence for some non - zero x in H we have Tx λα , and hence , since T = TE ...
... belongs to the spectrum of both T and ET . Suppose that λ 0 belongs to the spectrum of T. Since T is compact , Theorem VII.4.5 shows that 2 is an eigenvalue and hence for some non - zero x in H we have Tx λα , and hence , since T = TE ...
Page 1116
... belongs to the Hilbert - Schmidt class C2 . If we let Aq ; = V i 1 - p / 2 - = = q ;, then A is plainly self adjoint and A belongs to the class C ,, where r ( 1 - p / 2 ) = p , i.e. , r = p ( 1 - p / 2 ) -1 . Thus , by Lemma 9 , T BA ...
... belongs to the Hilbert - Schmidt class C2 . If we let Aq ; = V i 1 - p / 2 - = = q ;, then A is plainly self adjoint and A belongs to the class C ,, where r ( 1 - p / 2 ) = p , i.e. , r = p ( 1 - p / 2 ) -1 . Thus , by Lemma 9 , T BA ...
Page 1602
... belongs to the essential spectrum of 7 ( Hartman and Wintner [ 14 ] ) . ( 48 ) Suppose that the function q is bounded below , and let ƒ be a real solution of the equation ( 2-7 ) = 0 on [ 0 , ∞ ) which is not square - integrable but ...
... belongs to the essential spectrum of 7 ( Hartman and Wintner [ 14 ] ) . ( 48 ) Suppose that the function q is bounded below , and let ƒ be a real solution of the equation ( 2-7 ) = 0 on [ 0 , ∞ ) which is not square - integrable but ...
Contents
IX | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
Copyright | |
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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 compact subset complex numbers continuous function converges Corollary deficiency indices Definition denote dense domain 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 isometric isomorphism kernel L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood norm open set open subset orthonormal partial differential operator Plancherel's theorem positive PROOF prove real axis real numbers satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose T₁ T₁(t theory To(t topology unique unitary vanishes vector zero