Linear Operators: Spectral operators |
From inside the book
Results 1-3 of 73
Page 1027
... belongs to the spectrum of both T and ET . Suppose that 20 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 § we have Tx λυ , and hence , since T TE , we ...
... belongs to the spectrum of both T and ET . Suppose that 20 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 § we have Tx λυ , and hence , since T TE , we ...
Page 1116
... belongs to the Hilbert - Schmidt class C2 . If we let Ap1 = y / 2p , 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 ...
... belongs to the Hilbert - Schmidt class C2 . If we let Ap1 = y / 2p , 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 ...
Page 1684
... belongs to L „ ( E ) , it follows that every partial derivative of F of order not more than m is continuous in the closure of E. PROOF . By Corollary 2 and Hölder's inequality , each ( k - m ) th derivative of any Ith derivative of F ...
... belongs to L „ ( E ) , it follows that every partial derivative of F of order not more than m is continuous in the closure of E. PROOF . By Corollary 2 and Hölder's inequality , each ( k - m ) th derivative of any Ith derivative of F ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
Copyright | |
36 other sections not shown
Other editions - View all
Common terms and phrases
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 Haar measure 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 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 unique unitary vanishes vector zero