Linear Operators: Spectral operators |
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Page 1454
... equal . PROOF . To prove ( a ) , note that if T is bounded below , there exists a constant K such that ( Tx , x ) -K ( x , x ) , x = D ( T ) . The proof of Theorem 5 now shows that σ , ( T ) is a subset of the half- axis oot -K . Since ...
... equal . PROOF . To prove ( a ) , note that if T is bounded below , there exists a constant K such that ( Tx , x ) -K ( x , x ) , x = D ( T ) . The proof of Theorem 5 now shows that σ , ( T ) is a subset of the half- axis oot -K . Since ...
Page 1539
... equal deficiency indices , and let 2 be a real number . Prove that the distance from 2 to the essential spectrum of t is less than or equal to K if and only if there exists a sequence fn in D ( To ( T ) ) such that f2 = 1 , fn vanishes ...
... equal deficiency indices , and let 2 be a real number . Prove that the distance from 2 to the essential spectrum of t is less than or equal to K if and only if there exists a sequence fn in D ( To ( T ) ) such that f2 = 1 , fn vanishes ...
Page 1735
... equal to 1 in a neighborhood of p = 0 and identically equal to zero outside the unit sphere in E " . Let § in Co ° ( E " ) be identically equal to 1 in a neighborhood of the unit closed sphere in E and identically zero outside the ...
... equal to 1 in a neighborhood of p = 0 and identically equal to zero outside the unit sphere in E " . Let § in Co ° ( E " ) be identically equal to 1 in a neighborhood of the unit closed sphere in E and identically zero outside the ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 Haar measure 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 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 unique unitary vanishes vector zero