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 a we see , from Theorem 2.6 ( d ) , that ( F ( T ) x , y ) = ( x , F ( T ) y ) = 0 so that F ( T ) ...
... 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 a we see , from Theorem 2.6 ( d ) , that ( F ( T ) x , y ) = ( x , F ( T ) y ) = 0 so that F ( T ) ...
Page 1227
... orthogonal sub- spaces of the Hilbert space D ( T * ) . = ( b ) D ( T * ) = D ( T ) ✪ D , ✪ D_ . + + PROOF . By Lemma 8 ( a ) , D ( T ) is closed . Suppose { x } is a se- quence of elements of D converging to x = D ( T * ) , then ...
... orthogonal sub- spaces of the Hilbert space D ( T * ) . = ( b ) D ( T * ) = D ( T ) ✪ D , ✪ D_ . + + PROOF . By Lemma 8 ( a ) , D ( T ) is closed . Suppose { x } is a se- quence of elements of D converging to x = D ( T * ) , then ...
Page 1777
... orthogonal . An orthonormal set is said to be complete if no non - zero vector is orthogonal to every vector in the set , i.e. , A is complete if { 0 } = A. We recall that a projection is a linear operator E with E2 E. A projection E in ...
... orthogonal . An orthonormal set is said to be complete if no non - zero vector is orthogonal to every vector in the set , i.e. , A is complete if { 0 } = A. We recall that a projection is a linear operator E with E2 E. A projection E in ...
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