Linear Operators: Spectral operators |
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Page 1226
... follows immediately from part ( a ) and Lemma 5 ( c ) . Q.E.D. It follows from Lemma 6 ( b ) that any symmetric operator with dense domain has a unique minimal closed symmetric extension . This fact leads us to make the following ...
... follows immediately from part ( a ) and Lemma 5 ( c ) . Q.E.D. It follows from Lemma 6 ( b ) that any symmetric operator with dense domain has a unique minimal closed symmetric extension . This fact leads us to make the following ...
Page 1469
... follows immediately from Theorems 4.1 , 4.2 , and XII.7.2 that ( tf , f ) = ( Tcf , f ) ≥ ( λ 。— ε / 2 ) | f | 2 ... follows immediately from the preceding lemma . To prove statement ( b ) , note that is finite below 2 , but not below ...
... follows immediately from Theorems 4.1 , 4.2 , and XII.7.2 that ( tf , f ) = ( Tcf , f ) ≥ ( λ 。— ε / 2 ) | f | 2 ... follows immediately from the preceding lemma . To prove statement ( b ) , note that is finite below 2 , but not below ...
Page 1708
... follows that there exists some F in H ( + ) such that ( T1 + 0 ) F = ge ... immediately from Theorem 2 , Sobolev's theorem ( 4.5 ) , and the closed ... follows immediately from the preceding corollary and the remark immediately following ...
... follows that there exists some F in H ( + ) such that ( T1 + 0 ) F = ge ... immediately from Theorem 2 , Sobolev's theorem ( 4.5 ) , and the closed ... follows immediately from the preceding corollary and the remark immediately following ...
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 linear 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 satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology unique unitary vanishes vector zero