Linear Operators: Spectral theory |
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Page 1463
... zero in [ c , d ] , so that f1fz1 is constant . Moreover , since f1 and f1 have only a finite number of zeros in [ c ... zero , there exists a zero of any linearly independent solution f2 ; τή = ( b ) if any solution of τf1 = 0 which is ...
... zero in [ c , d ] , so that f1fz1 is constant . Moreover , since f1 and f1 have only a finite number of zeros in [ c ... zero , there exists a zero of any linearly independent solution f2 ; τή = ( b ) if any solution of τf1 = 0 which is ...
Page 1474
... zero between every pair of zeros of σ ( t , λ1 ) , we have only to show that the interval ( a , z ] between a and the smallest zero z of o ( t , λ ) contains a zero of σ ( t , λ ) , and we will have established that σ ( t , λ ) has at ...
... zero between every pair of zeros of σ ( t , λ1 ) , we have only to show that the interval ( a , z ] between a and the smallest zero z of o ( t , λ ) contains a zero of σ ( t , λ ) , and we will have established that σ ( t , λ ) has at ...
Page 1475
... zero in ( a , z1 ] and a zero in [ zą , b ) , we will have established that σ ( · , λ ) has at least n + 1 zeros in ( a , b ) , contradicting the fact that λ is in J. It is sufficient to prove that σ ( ' , λ2 ) has a zero in ( a , z1 ] ...
... zero in ( a , z1 ] and a zero in [ zą , b ) , we will have established that σ ( · , λ ) has at least n + 1 zeros in ( a , b ) , contradicting the fact that λ is in J. It is sufficient to prove that σ ( ' , λ2 ) has a zero in ( a , z1 ] ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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Common terms and phrases
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 Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set 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 unique unitary vanishes vector zero