Linear Operators: Spectral theory |
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Page 1650
... subset I of E " . Then the closed set C in I which is the complement in I of the largest open set in I in which F vanishes , i.e. , which is the complement in I of the union of all the open subsets of I in which F vanishes , is called ...
... subset I of E " . Then the closed set C in I which is the complement in I of the largest open set in I in which F vanishes , i.e. , which is the complement in I of the union of all the open subsets of I in which F vanishes , is called ...
Page 1663
... subset I。 of I whose closure is compact and contained in I will be denoted by A * ) ( I ) . each 36 DEFINITION . Let I be an open subset of C. Let k be an integer , positive or negative . Let { I } , m ≥1 , be a sequence of open subsets ...
... subset I。 of I whose closure is compact and contained in I will be denoted by A * ) ( I ) . each 36 DEFINITION . Let I be an open subset of C. Let k be an integer , positive or negative . Let { I } , m ≥1 , be a sequence of open subsets ...
Page 1695
... subset I of E " . Let { I } be a sequence of open subsets of I whose union is I , such that Im is compact and contained in I , and such that ImÏ1 = │m - p│ 1. Then F may be written as the sum F convergent infinite series of ...
... subset I of E " . Let { I } be a sequence of open subsets of I whose union is I , such that Im is compact and contained in I , and such that ImÏ1 = │m - p│ 1. Then F may be written as the sum F convergent infinite series of ...
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