Linear Operators: Spectral theory |
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Page 1187
... closed operator is closed . A bounded operator is closed if and only if its domain is closed . - PROOF . If A is the isometric automorphism in which maps [ x , y ] into [ y , x ] then П ( T − 1 ) = Â ̧Ã ( T ) which shows that T is ...
... closed operator is closed . A bounded operator is closed if and only if its domain is closed . - PROOF . If A is the isometric automorphism in which maps [ x , y ] into [ y , x ] then П ( T − 1 ) = Â ̧Ã ( T ) which shows that T is ...
Page 1393
... closed operator in Hilbert space . Then the set of complex numbers & such that the range of I —T is not closed is called the essential spectrum of T and is denoted by σ ( T ) . It is clear that σ , ( T ) ≤ o ( T ) . If 7 is a formal ...
... closed operator in Hilbert space . Then the set of complex numbers & such that the range of I —T is not closed is called the essential spectrum of T and is denoted by σ ( T ) . It is clear that σ , ( T ) ≤ o ( T ) . If 7 is a formal ...
Page 1394
... closed . Hence we may assume that Ta & TY . Suppose that TY is not closed , and let z ‡ TY , ZETY , so that there exists a sequence of elements y1 = Y such that Tyz . Since T ( Y + N ) is closed , there exists an element y + ar , ye Y ...
... closed . Hence we may assume that Ta & TY . Suppose that TY is not closed , and let z ‡ TY , ZETY , so that there exists a sequence of elements y1 = Y such that Tyz . Since T ( Y + N ) is closed , there exists an element y + ar , ye Y ...
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