Linear Operators: Spectral theory |
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Page 1088
... operator in L2 ( S , Σ , μ ) . Let { 9 } be an enumeration of the eigenfunctions of K , and { u } an enumeration of the correspond- ing eigenvalues . Show that if g Kf for some f in L ... Compact Operators Generalized Carleman Inequalities.
... operator in L2 ( S , Σ , μ ) . Let { 9 } be an enumeration of the eigenfunctions of K , and { u } an enumeration of the correspond- ing eigenvalues . Show that if g Kf for some f in L ... Compact Operators Generalized Carleman Inequalities.
Page 1089
... compact or non- compact operator satisfy the inequalities Mn + m + 1 ( T1 + T2 ) ≤ μn + 1 ( T1 ) + μm + 1 ( T2 ) Un + m + 1 ( TiT , ) Sun + 1 ( Ti ) um + 1 ( T2 ) . PROOF . We observe that min P1 , ... , XI.9.1 1089 CLASSES OF COMPACT ...
... compact or non- compact operator satisfy the inequalities Mn + m + 1 ( T1 + T2 ) ≤ μn + 1 ( T1 ) + μm + 1 ( T2 ) Un + m + 1 ( TiT , ) Sun + 1 ( Ti ) um + 1 ( T2 ) . PROOF . We observe that min P1 , ... , XI.9.1 1089 CLASSES OF COMPACT ...
Page 1095
... operators such that T - Tm0 as m , n∞ , there exists a compact operator T such that TT ( in the topology of C ) as n∞ . n PROOF . By Lemma 9 ( a ) and the fact that the family of compact operators is closed in the uniform topology of ...
... operators such that T - Tm0 as m , n∞ , there exists a compact operator T such that TT ( in the topology of C ) as n∞ . n PROOF . By Lemma 9 ( a ) and the fact that the family of compact operators is closed in the uniform topology of ...
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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