Linear Operators: Spectral theory |
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Page 972
... equal to unity , it follows from Plan- cherel's theorem that { u ( e + p ) } 2 = { u ( e ) } 2 . Hence if μ ( e ) < ∞ , we have proved that μ ( e + p ) is also finite and equals u ( e ) . If u ( e + p ) were known to be finite we would ...
... equal to unity , it follows from Plan- cherel's theorem that { u ( e + p ) } 2 = { u ( e ) } 2 . Hence if μ ( e ) < ∞ , we have proved that μ ( e + p ) is also finite and equals u ( e ) . If u ( e + p ) were known to be finite we would ...
Page 1454
... equal . PROOF . To prove ( a ) , note that if T is bounded below , there exists a constant K such that ( Tx , x ) ≥ K ( x , x ) , x Є D ( T ) . The proof of Theorem 5 now shows that σ ( T ) is a subset of the half- axis oot - K . Since ...
... equal . PROOF . To prove ( a ) , note that if T is bounded below , there exists a constant K such that ( Tx , x ) ≥ K ( x , x ) , x Є D ( T ) . The proof of Theorem 5 now shows that σ ( T ) is a subset of the half- axis oot - K . Since ...
Page 1539
... equal deficiency indices , and let λ be a real number . Prove that the distance from 2 to the essential spectrum of t is less than or equal to K if and only if there exists a sequence ƒ „ in D ( To ( T ) ) such that f = 1 , f vanishes ...
... equal deficiency indices , and let λ be a real number . Prove that the distance from 2 to the essential spectrum of t is less than or equal to K if and only if there exists a sequence ƒ „ in D ( To ( T ) ) such that f = 1 , f vanishes ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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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 complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero