Linear Operators: Spectral theory |
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Page 1088
... operator in L2 ( S , E , μ ) . Let { 9 } be an enumeration of the eigenfunctions of K , and { μ } an enumeration of the correspond- ing eigenvalues . Show that if g = Kf for some f in ... Compact Operators Generalized Carleman Inequalities.
... operator in L2 ( S , E , μ ) . Let { 9 } be an enumeration of the eigenfunctions of K , and { μ } an enumeration of the correspond- ing eigenvalues . Show that if g = Kf for some f in ... Compact Operators Generalized Carleman Inequalities.
Page 1089
... compact or non- compact operator satisfy the inequalities Un + m + 1 ( Ti + T , ) = Mn + 1 ( Ti ) + um + 1 ( T2 ) Un + m + 1 ( TiT , ) < Un + 1 ( Ti ) m + 1 ( T ) . PROOF . We observe that min P1 , ... , XI.9.1 1089 CLASSES OF COMPACT ...
... compact or non- compact operator satisfy the inequalities Un + m + 1 ( Ti + T , ) = Mn + 1 ( Ti ) + um + 1 ( T2 ) Un + m + 1 ( TiT , ) < Un + 1 ( Ti ) m + 1 ( T ) . PROOF . We observe that min P1 , ... , XI.9.1 1089 CLASSES OF COMPACT ...
Page 1095
... operators such that T2 - Tmp → 0 as m , n → ∞ , there exists a compact operator T such that TT ( in the topology of C1 ) as n → ∞ . PROOF . By Lemma 9 ( a ) and the fact that the family of compact operators is closed in the uniform ...
... operators such that T2 - Tmp → 0 as m , n → ∞ , there exists a compact operator T such that TT ( in the topology of C1 ) as n → ∞ . PROOF . By Lemma 9 ( a ) and the fact that the family of compact operators is closed in the uniform ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero