Linear Operators: Spectral theory |
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Page 1105
... Lemma 13 ( b ) . = We now pause to sharpen another of the inequalities of Lemma 9 . 20 LEMMA . Let A1 € C ,,, A2 € С ,,, A ̧¤ С ,, where rï1 + rğ1 + rğ1 1. Then ( a ) | │tr ( 41243 ) | ≤ | 41 | r , | 42 | r2 | 43 | 2 ( b ) if r1 = rī1 ...
... Lemma 13 ( b ) . = We now pause to sharpen another of the inequalities of Lemma 9 . 20 LEMMA . Let A1 € C ,,, A2 € С ,,, A ̧¤ С ,, where rï1 + rğ1 + rğ1 1. Then ( a ) | │tr ( 41243 ) | ≤ | 41 | r , | 42 | r2 | 43 | 2 ( b ) if r1 = rī1 ...
Page 1226
... Lemma 5 ( c ) . Q.E.D. It follows from Lemma 6 ( b ) that any symmetric operator with dense domain has a unique minimal closed symmetric extension . This fact leads us to make the following definition . 7 DEFINITION . The minimal closed ...
... Lemma 5 ( c ) . Q.E.D. It follows from Lemma 6 ( b ) that any symmetric operator with dense domain has a unique minimal closed symmetric extension . This fact leads us to make the following definition . 7 DEFINITION . The minimal closed ...
Page 1696
... lemma , let m be a sequence in C ( D ) with m → G in the topology of H ( * ) ( D ) as m → ∞ . Using Lemma 2.1 , let the function y in Co ( I ) be such that y ( x ) = 1 for all x in a neighborhood of C. Then , by Lemma 3.10 and ...
... lemma , let m be a sequence in C ( D ) with m → G in the topology of H ( * ) ( D ) as m → ∞ . Using Lemma 2.1 , let the function y in Co ( I ) be such that y ( x ) = 1 for all x in a neighborhood of C. Then , by Lemma 3.10 and ...
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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