Linear Operators, Part 2 |
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Page 972
... equal to unity , it follows from Plan- cherel's theorem that { μ ( e + p ) } 2 = { μ ( e ) } 2 . Hence if μ ( e ) < ∞ , we have proved that μ ( e + p ) is also finite and equals μ ( e ) . If μ ( e + p ) were known to be finite we would ...
... equal to unity , it follows from Plan- cherel's theorem that { μ ( e + p ) } 2 = { μ ( e ) } 2 . Hence if μ ( e ) < ∞ , we have proved that μ ( e + p ) is also finite and equals μ ( e ) . If μ ( e + p ) were known to be finite we would ...
Page 1231
... equal deficiency indices , and thus has self adjoint extensions . 17 DEFINITION . A mapping U of S into itself which satisfies U ( x + y ) = Ux + Uy , U ( xx ) = ăUx , ( Ux , Uy ) = ( y , x ) , x , y ɛ H , and U2 = 1 , is called a ...
... equal deficiency indices , and thus has self adjoint extensions . 17 DEFINITION . A mapping U of S into itself which satisfies U ( x + y ) = Ux + Uy , U ( xx ) = ăUx , ( Ux , Uy ) = ( y , x ) , x , y ɛ H , and U2 = 1 , is called a ...
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 . 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 = D ( T ) . The proof of Theorem 5 now shows that σ ( T ) is a subset of the half- axis > t -- K . Since ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 coefficients compact operator complex numbers constant 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 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 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 subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero