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 , . . . , ƒ ” + 1 in t be defined by the equations fl = [ Xo , 0 , 0 , ... ] , = [ 0 , Xo , 0 , ... ] , fn + 1 = [ 0 , ... , 0 , % , 0 ...
... arbitrary fixed Borel subset of en + 1 - en + 1 such that 0 < μ ( o ) < ∞o . Let the vectors ƒ1 , . . . , ƒ ” + 1 in t be defined by the equations fl = [ Xo , 0 , 0 , ... ] , = [ 0 , Xo , 0 , ... ] , fn + 1 = [ 0 , ... , 0 , % , 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 , ɛ ) 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 , ɛ ) are a base for a topology will be left to the reader . If h1 = N ...
Page 1337
... arbitrary vector f in L2 ( I ) has an expansion of " Fourier integral " type in terms of eigenfunctions W ( t , 2 ) of the differential operator v . Unfortunately , the interest of Theorem 1 is more theoretical than practical , since it ...
... arbitrary vector f in L2 ( I ) has an expansion of " Fourier integral " type in terms of eigenfunctions W ( t , 2 ) of the differential operator v . Unfortunately , the interest of Theorem 1 is more theoretical than practical , since it ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero