Linear Operators: Spectral theory |
From inside the book
Results 1-3 of 73
Page 1083
... Exercise 43 is the space L¿ ( S , Σ , 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 ∞ 4 ( s , t ; 2 ) = Σλη Δη ...
... Exercise 43 is the space L¿ ( S , Σ , 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 ∞ 4 ( s , t ; 2 ) = Σλη Δη ...
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 p 50 ( 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. Miscellaneous Exercises p 50 ( 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 ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
Copyright | |
45 other sections not shown
Other editions - View all
Common terms and phrases
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