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 u ( e ) < ∞ , we have proved that μ ( e + p ) is also finite and equals u ( e ) . If μ ( 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 u ( e ) < ∞ , we have proved that μ ( e + p ) is also finite and equals u ( e ) . If μ ( 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 = D ( T ) . The proof of Theorem 5 now shows that σ , ( T ) is a subset of the half- axis > t - K ...
... 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 = D ( T ) . The proof of Theorem 5 now shows that σ , ( T ) is a subset of the half- axis > t - K ...
Page 1539
... equal deficiency indices , and let λ be a real number . Prove that the distance from λ 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 ( 7 ) ) such that f 1 , fn vanishes on ...
... equal deficiency indices , and let λ be a real number . Prove that the distance from λ 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 ( 7 ) ) such that f 1 , fn vanishes on ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero