Linear Operators: Spectral theory |
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Page 1401
... mapping → x1 of N into I maps N onto x1 all of T. By the two preceding lemmas , N is exactly k - dimensional ; because I is also k - dimensional , the mapping xx1 is one - to - one . Furthermore , I is the graph of an isometry between ...
... mapping → x1 of N into I maps N onto x1 all of T. By the two preceding lemmas , N is exactly k - dimensional ; because I is also k - dimensional , the mapping xx1 is one - to - one . Furthermore , I is the graph of an isometry between ...
Page 1669
... mapping → M is a continuous mapping of C∞ ( 12 ) into C∞ ( I1 ) . 9 ( See Section 2 for a definition of the topology in these spaces . ) By ( a ) , poM maps Co ( I2 ) into Co ( I ) . By ( a ) again , all the functions of the sequence ...
... mapping → M is a continuous mapping of C∞ ( 12 ) into C∞ ( I1 ) . 9 ( See Section 2 for a definition of the topology in these spaces . ) By ( a ) , poM maps Co ( I2 ) into Co ( I ) . By ( a ) again , all the functions of the sequence ...
Page 1671
... mapping → M - 1 ( x ) ; this follows by the standard theorem on change of variables in a multiple integral . But then ( iii ) is evident . Q.E.D. Lemma 47 allows us to describe the behavior of the spaces H " , AP , etc. , under the ...
... mapping → M - 1 ( x ) ; this follows by the standard theorem on change of variables in a multiple integral . But then ( iii ) is evident . Q.E.D. Lemma 47 allows us to describe the behavior of the spaces H " , AP , etc. , under the ...
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