Linear Operators, Part 2 |
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Page 918
... arbitrary fixed Borel subset of en + 1 - ễn + 1 such that 0 < μ ( o ) < ∞ . Let the vectors ƒ1 , . . . , ƒ ” + 1 in § be defined by the equations f1 = [ Xo , 0 , 0 , ... ] , f2 = [ 0 , Xo , 0 , . . . ] , fn + 1 = [ 0 , ... , 0 , % , 0 ...
... arbitrary fixed Borel subset of en + 1 - ễn + 1 such that 0 < μ ( o ) < ∞ . Let the vectors ƒ1 , . . . , ƒ ” + 1 in § be defined by the equations f1 = [ Xo , 0 , 0 , ... ] , f2 = [ 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 , 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 L2 ( I ) has an expansion of " Fourier integral " type in terms of eigenfunctions W , ( t , 2 ) of the differential operator 7. Unfortunately , the interest of Theorem 1 is more theoretical than practical , since ...
... arbitrary vector f in L2 ( I ) has an expansion of " Fourier integral " type in terms of eigenfunctions W , ( t , 2 ) of the differential operator 7. Unfortunately , the interest of Theorem 1 is more theoretical than practical , since ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 coefficients compact operator complex numbers constant 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 subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero