Linear Operators: Spectral theory |
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Page 970
... limit in the norm of L2 ( R ) of the generalized sequence { xef } . Hence , by Theorem 9 , tf is the limit in the norm of L2 ( M ) of the generalized sequence { T ( ef ) } . Equivalently , we write τf С = lim [ x , f ( x ) dx , e e ...
... limit in the norm of L2 ( R ) of the generalized sequence { xef } . Hence , by Theorem 9 , tf is the limit in the norm of L2 ( M ) of the generalized sequence { T ( ef ) } . Equivalently , we write τf С = lim [ x , f ( x ) dx , e e ...
Page 1124
... limit q ( E ) , then it follows from what we have already proved that E , is an increasing sequence of projections and EE . If E is the strong limit of E ,,, then E. E and ( E ) = q ( E ) . Thus , it follows as above that E E. This ...
... limit q ( E ) , then it follows from what we have already proved that E , is an increasing sequence of projections and EE . If E is the strong limit of E ,,, then E. E and ( E ) = q ( E ) . Thus , it follows as above that E E. This ...
Page 1699
... limit in the norm of HP ) ( C ) of the sequence { g } of elements of Co ( C ) . Hence , by Lemma 3.23 , ( F ) \ C + = q ( Fot1 ) | C , is the limit in the norm of HP ) ( C ) of the sequence { g , C } of functions . It then follows from ...
... limit in the norm of HP ) ( C ) of the sequence { g } of elements of Co ( C ) . Hence , by Lemma 3.23 , ( F ) \ C + = q ( Fot1 ) | C , is the limit in the norm of HP ) ( C ) of the sequence { g , C } of functions . It then follows from ...
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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