Linear Operators: Spectral theory |
From inside the book
Results 1-3 of 75
Page 1083
... Exercise 43 is the space L2 ( S , 2 , μ ) of Exercise 44. Let 4 „ ( s , t ) be the kernel of Exercise 44 which represents , in the sense of Exercise 44 , the operator 4 , of Exercise 42. Then the power series △ ( s , t ; 2 ) = Σ 2 ...
... Exercise 43 is the space L2 ( S , 2 , μ ) of Exercise 44. Let 4 „ ( s , t ) be the kernel of Exercise 44 which represents , in the sense of Exercise 44 , the operator 4 , of Exercise 42. Then the power series △ ( 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 ...
... 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 ...
Page 1087
... Exercise 30. ) E. Miscellaneous Exercises Р 50 ( Halberg ) Let ( S , Σ , μ ) be a σ - finite measure space . Let T2 be a 1 - parameter family of bounded operators defined in a sub- interval I of the parameter interval 1 ≤ p ≤∞ , each ...
... Exercise 30. ) E. Miscellaneous Exercises Р 50 ( Halberg ) Let ( S , Σ , μ ) be a σ - finite measure space . Let T2 be a 1 - parameter family of bounded operators defined in a sub- interval I of the parameter interval 1 ≤ p ≤∞ , each ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
Copyright | |
52 other sections not shown
Other editions - View all
Common terms and phrases
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 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 PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero