Linear Operators: Spectral theory |
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Page 1083
... Exercise 43 is the space L2 ( S , E , μ ) of Exercise 44. Let 4n ( s , t ) be the kernel of Exercise 44 which represents , in the sense of Exercise 44 , the operator 4 Exercise 42. Then the power series ∞ 4 ( s , t ; 2 ) = Σ 2 ...
... Exercise 43 is the space L2 ( S , E , μ ) of Exercise 44. Let 4n ( s , t ) be the kernel of Exercise 44 which represents , in the sense of Exercise 44 , the operator 4 Exercise 42. Then the power series ∞ 4 ( s , t ; 2 ) = Σ 2 ...
Page 1086
... exercise , in the sense of Exercise 44 . Show , finally , that by choosing A ( s , s ) = 0 for all s in S , we obtain the result of Exercise 46 as a special case of the present result . ( Hint : Generalize the method of Exercise 46. ) i ...
... exercise , in the sense of Exercise 44 . Show , finally , that by choosing A ( s , s ) = 0 for all s in S , we obtain the result of Exercise 46 as a special case of the present result . ( Hint : Generalize the method of Exercise 46. ) i ...
Page 1087
... Exercise 30. ) E. 50 Miscellaneous Exercises p ( Halberg ) Let ( S , Σ , μ ) be a σ - finite measure space . Let T be a 1 - parameter family of bounded operators defined in a sub- interval I of the parameter interval 1 ≤ p ≤∞ , each ...
... Exercise 30. ) E. 50 Miscellaneous Exercises p ( Halberg ) Let ( S , Σ , μ ) be a σ - finite measure space . Let T be a 1 - parameter family of bounded operators defined in a sub- interval I of the parameter interval 1 ≤ p ≤∞ , each ...
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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