Linear Operators: Spectral theory |
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Page 918
... arbitrary fixed Borel subset of en + 1 - en + 1 such that 0 < μ ( o ) < ∞o . Let the vectors ƒ1 , . fn + 1 in be defined by the equations • " fi f2 = = [ Xo , 0 , 0 , ... ] , [ 0 , Xo , 0 , ... ] , fn + 1 = [ 0 , ... , 0 , Xo , 0 ...
... arbitrary fixed Borel subset of en + 1 - en + 1 such that 0 < μ ( o ) < ∞o . Let the vectors ƒ1 , . fn + 1 in be defined by the equations • " fi f2 = = [ Xo , 0 , 0 , ... ] , [ 0 , Xo , 0 , ... ] , fn + 1 = [ 0 , ... , 0 , Xo , 0 ...
Page 968
... arbitrary and K is an arbitrary compact subset of R. 14 LEMMA . The character group R is a topological group . PROOF . Verification that the neighborhoods N ( h , K , e ) are a base for a topology will be left to the reader . If h1 = N ...
... arbitrary and K is an arbitrary compact subset of R. 14 LEMMA . The character group R is a topological group . PROOF . Verification that the neighborhoods N ( h , K , e ) are a base for a topology will be left to the reader . If h1 = N ...
Page 1337
... arbitrary vector f in La ( I ) has an expansion of " Fourier integral " type in terms of eigenfunctions W ( t , 2 ) of the differential operator t . Unfortunately , the interest of Theorem 1 is more theoretical than practical , since it ...
... arbitrary vector f in La ( I ) has an expansion of " Fourier integral " type in terms of eigenfunctions W ( t , 2 ) of the differential operator t . Unfortunately , the interest of Theorem 1 is more theoretical than practical , since it ...
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