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Page 861
... exists then TT TT , 1 = I. If T exists in B ( X ) , then x x T ̧ [ ( T71y ) z ] = yz , x x ( T1y ) z = Tz1 ( yz ) , and if a = T1e , then az = Tz for every ze X. Also xa = T2a = e = T - 1 ( T ̧e ) = Tz1 ( ex ) = ( Tz1e ) x = ax . Thus 1 ...
... exists then TT TT , 1 = I. If T exists in B ( X ) , then x x T ̧ [ ( T71y ) z ] = yz , x x ( T1y ) z = Tz1 ( yz ) , and if a = T1e , then az = Tz for every ze X. Also xa = T2a = e = T - 1 ( T ̧e ) = Tz1 ( ex ) = ( Tz1e ) x = ax . Thus 1 ...
Page 1057
... exists and t > 0 ; and the integral ( Vu ) exists and equals Ω ( α ) Pa ei ( x , Vu ) dx = P S Q ( Vy ) ei ( v , u ) dy Ey " En if PSgn ( Vy ) yn ei ( v , u ) dy exists and V is a rotation of E " . Thus , to show that the proper value ...
... exists and t > 0 ; and the integral ( Vu ) exists and equals Ω ( α ) Pa ei ( x , Vu ) dx = P S Q ( Vy ) ei ( v , u ) dy Ey " En if PSgn ( Vy ) yn ei ( v , u ) dy exists and V is a rotation of E " . Thus , to show that the proper value ...
Page 1262
... exists a Hilbert space 125 , and an orthogonal projection Q in such that Ax = PQx , x = H , P denoting the orthogonal projection of 1 on H. 29 Let { T } be a sequence of bounded operators in Hilbert space . Then there exists a Hilbert ...
... exists a Hilbert space 125 , and an orthogonal projection Q in such that Ax = PQx , x = H , P denoting the orthogonal projection of 1 on H. 29 Let { T } be a sequence of bounded operators in Hilbert space . Then there exists a Hilbert ...
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