Linear Operators: Spectral theory |
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Page 1002
... Show that a ( 2 ) except for at most a countable infinity of values λ , i = 1 , 2 , ... , and that M ( g ) = Σ1 | a ( 2 ; ) | 2 . 4 If ƒ is a non - negative function in AP , and M ( ƒ ) notation of Exercise 2 ) then f = 0 . = 0 ( in the ...
... Show that a ( 2 ) except for at most a countable infinity of values λ , i = 1 , 2 , ... , and that M ( g ) = Σ1 | a ( 2 ; ) | 2 . 4 If ƒ is a non - negative function in AP , and M ( ƒ ) notation of Exercise 2 ) then f = 0 . = 0 ( in the ...
Page 1074
... Show that + A F ( t ) = lim eitz f ( x ) dx A → ∞ A -1 = 1. ( Hint : exists in the norm of L ( -∞ , ∞ ) , where p1 + q - 1 Cf. VI.11.43 . ) 7 Show , with the hypotheses and notation of Exercise 6 , that lim A → ∞ + A 1 [ ** F ( t ) ...
... Show that + A F ( t ) = lim eitz f ( x ) dx A → ∞ A -1 = 1. ( Hint : exists in the norm of L ( -∞ , ∞ ) , where p1 + q - 1 Cf. VI.11.43 . ) 7 Show , with the hypotheses and notation of Exercise 6 , that lim A → ∞ + A 1 [ ** F ( t ) ...
Page 1548
... Show that the operator T is self adjoint . Show that the operator T is bounded below if and only if both T1 and T2 are bounded below . Let λn ( T1 ) , λn ( T2 ) , and λ „ ( T ) be the numbers defined in Exercise D2 , for the operators ...
... Show that the operator T is self adjoint . Show that the operator T is bounded below if and only if both T1 and T2 are bounded below . Let λn ( T1 ) , λn ( T2 ) , and λ „ ( T ) be the numbers defined in Exercise D2 , for the operators ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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