Linear Operators: Spectral theory |
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Page 1463
... Statement ( a ) is the special case t1 = T2 of the preceding theorem . Statement ( b ) is an evident consequence of statement ( a ) . Q.E.D. The theorem is illustrated by the pair of functions sin λt and sin ut for u > 2 ; the corollary ...
... Statement ( a ) is the special case t1 = T2 of the preceding theorem . Statement ( b ) is an evident consequence of statement ( a ) . Q.E.D. The theorem is illustrated by the pair of functions sin λt and sin ut for u > 2 ; the corollary ...
Page 1469
... statement ( a ) follows immediately from the preceding lemma . To prove statement ( b ) , note that is finite below 2 , but not below any > 2. Statement ( b ) then follows immediately from the preceding lemma . Q.E.D. Next we turn to ...
... statement ( a ) follows immediately from the preceding lemma . To prove statement ( b ) , note that is finite below 2 , but not below any > 2. Statement ( b ) then follows immediately from the preceding lemma . Q.E.D. Next we turn to ...
Page 1771
... Statement ( i ) follows from the preceding theorem and Theorem 6.23 . Statement ( ii ) follows from statement ( ii ) of the preceding theorem , since a function satisfying the hypotheses of the present statement ( ii ) evidently ( cf ...
... Statement ( i ) follows from the preceding theorem and Theorem 6.23 . Statement ( ii ) follows from statement ( ii ) of the preceding theorem , since a function satisfying the hypotheses of the present statement ( ii ) evidently ( cf ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero