Linear Operators: Spectral theory |
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Page 1207
... orthogonal to each of the spaces H. Indeed , if x 0 is orthogonal to the space Ha then , for a bounded Borel function F and a point y in S , we see , from Theorem 2.6 ( d ) , that ( F ( T ) x , y ) = ( x , F ( T ) y ) ( x , F ( T ) y ) ...
... orthogonal to each of the spaces H. Indeed , if x 0 is orthogonal to the space Ha then , for a bounded Borel function F and a point y in S , we see , from Theorem 2.6 ( d ) , that ( F ( T ) x , y ) = ( x , F ( T ) y ) ( x , F ( T ) y ) ...
Page 1227
... orthogonal sub- spaces of the Hilbert space D ( T * ) . ( b ) D ( T ) = D ( T ) → D , ✪ D_ . + D + PROOF . By Lemma 8 ( a ) , D ( T ) is closed . Suppose { x } is a se- quence of elements of D converging to a = D ( T * ) , then { [ x ...
... orthogonal sub- spaces of the Hilbert space D ( T * ) . ( b ) D ( T ) = D ( T ) → D , ✪ D_ . + D + PROOF . By Lemma 8 ( a ) , D ( T ) is closed . Suppose { x } is a se- quence of elements of D converging to a = D ( T * ) , then { [ x ...
Page 1262
... orthogonal projection Q in S , such that Ax PQx , x = H , = P denoting the orthogonal projection of H1 on H. Let { T } be a sequence of bounded operators in Hilbert space H. Then there exists a Hilbert space 12H , and a sequence { N } ...
... orthogonal projection Q in S , such that Ax PQx , x = H , = P denoting the orthogonal projection of H1 on H. Let { T } be a sequence of bounded operators in Hilbert space H. Then there exists a Hilbert space 12H , and a sequence { N } ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero