Linear Operators: Spectral theory |
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Page 1452
... evident from Definition XII.5.1 that if T2 is bounded below , T1 must also be bounded below . Conversely , let T1 be bounded below . Suppose that the lemma is false , so that T2 is not bounded below . 2 We shall show by induction that ...
... evident from Definition XII.5.1 that if T2 is bounded below , T1 must also be bounded below . Conversely , let T1 be bounded below . Suppose that the lemma is false , so that T2 is not bounded below . 2 We shall show by induction that ...
Page 1631
... evident that T is linear , and equally evident that T is closed . Hence , by the closed graph theorem ( II.2.4 ) , T is continuous . Hence given any ɛ > 0 and any k , there is an integer l and a 8 > 0 such that μ ( k , T [ go , · · · gm ...
... evident that T is linear , and equally evident that T is closed . Hence , by the closed graph theorem ( II.2.4 ) , T is continuous . Hence given any ɛ > 0 and any k , there is an integer l and a 8 > 0 such that μ ( k , T [ go , · · · gm ...
Page 1662
... evident consequence of Lemma 12 , as generalized to D , ( I ) , which will be of importance to us in sub- sequent discussions , is stated in the following lemma . 33 LEMMA . Let I be an open subset of the interior of C , and let F be in ...
... evident consequence of Lemma 12 , as generalized to D , ( I ) , which will be of importance to us in sub- sequent discussions , is stated in the following lemma . 33 LEMMA . Let I be an open subset of the interior of C , and let F be in ...
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 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