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 . n PROOF . If A1 is the isometric automorphism in H H which maps [ x , y ] into [ y , a ] then I ( T - 1 ) = A , ( T ) which shows that T is ...
... closed operator is closed . A bounded operator is closed if and only if its domain is closed . n PROOF . If A1 is the isometric automorphism in H H which maps [ x , y ] into [ y , a ] then I ( T - 1 ) = A , ( 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 o , ( 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 o , ( T ) ≤ o ( T ) . If 7 is a formal ...
Page 1394
... closed . Hence we may assume that Tx & TY . Suppose that TY is not closed , and let z & TY , ZETY , so that there exists a sequence of elements y , E such that Tyn → z . Since T ( Y + N ) is closed , there exists an element y + xx , ye ...
... closed . Hence we may assume that Tx & TY . Suppose that TY is not closed , and let z & TY , ZETY , so that there exists a sequence of elements y , E such that Tyn → z . Since T ( Y + N ) is closed , there exists an element y + xx , ye ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero