Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 861
... exists , then TT - 1 . Clearly if x - 1 exists then T - TTT - 1 = I. If T1 exists in B ( X ) , then and if a = T ̧ [ ( Tz1y ) z ] = yz , ( T71y ) z = T71 ( yz ) , T1e , then az = Tz for every ze X. Also xa = T1a = e = T21 ( T ̧e ) = T21 ...
... exists , then TT - 1 . Clearly if x - 1 exists then T - TTT - 1 = I. If T1 exists in B ( X ) , then and if a = T ̧ [ ( Tz1y ) z ] = yz , ( T71y ) z = T71 ( yz ) , T1e , then az = Tz for every ze X. Also xa = T1a = e = T21 ( T ̧e ) = T21 ...
Page 1057
... exists and t > 0 ; and the integral ( Vu ) exists and equals P Q ( x ) En xn Ω ( Vy ) ei ( x , Vu ) dx = P ei ( y , u ) dy En y " if Pgn ( Vy ) | y | " ei ( v , u ) dy exists and V is a rotation of E " . Thus , to show that the proper ...
... exists and t > 0 ; and the integral ( Vu ) exists and equals P Q ( x ) En xn Ω ( Vy ) ei ( x , Vu ) dx = P ei ( y , u ) dy En y " if Pgn ( Vy ) | y | " ei ( v , u ) dy exists and V is a rotation of E " . Thus , to show that the proper ...
Page 1261
... exists a unique closed linear extension T such that if T , is any closed linear extension of T then TCT1 . T is called the closure of T. ( a ) There exists a densely defined operator with no closed linear extension . ( b ) An operator T ...
... exists a unique closed linear extension T such that if T , is any closed linear extension of T then TCT1 . T is called the closure of T. ( a ) There exists a densely defined operator with no closed linear extension . ( b ) An operator T ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 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 real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero