Linear Operators, Part 2 |
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Page 1026
... fact that tr ( T , T * ) = || T || 2 follow immediately from Definitions 1 and 17. Q.E.D. It follows readily from the results stated above that HS forms a Hilbert space under the inner product [ S , T ] = tr ( S , T * ) . Although this fact ...
... fact that tr ( T , T * ) = || T || 2 follow immediately from Definitions 1 and 17. Q.E.D. It follows readily from the results stated above that HS forms a Hilbert space under the inner product [ S , T ] = tr ( S , T * ) . Although this fact ...
Page 1245
... fact that each complex number a has a unique representation a where r≥ 0 , and | ei | 1. By analogy with the fact that r = we shall first seek to obtain the self adjoint operator A from the operator T * T . - α = reio , ( ãα ) , 1 ...
... fact that each complex number a has a unique representation a where r≥ 0 , and | ei | 1. By analogy with the fact that r = we shall first seek to obtain the self adjoint operator A from the operator T * T . - α = reio , ( ãα ) , 1 ...
Page 1348
... fact makes evident what could readily be suspected before : that from the point of view of the negative real axis , a more favorable choice of basis for the solutions of to - λσ is = ô1 ( t , 2 ) = e − t√ — à ̧ ô1⁄2 ( t , λ ) = et ...
... fact makes evident what could readily be suspected before : that from the point of view of the negative real axis , a more favorable choice of basis for the solutions of to - λσ is = ô1 ( t , 2 ) = e − t√ — à ̧ ô1⁄2 ( t , λ ) = et ...
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 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 real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero