Linear Operators: Spectral theory |
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Page 1628
... differential operators of the second order can be given . To construct the operators T ( s ) , construct two solutions u and v of the partial ... operator T ( t ) will then satisfy conditions ( a ) through ( f ) . A detailed exposition of ...
... differential operators of the second order can be given . To construct the operators T ( s ) , construct two solutions u and v of the partial ... operator T ( t ) will then satisfy conditions ( a ) through ( f ) . A detailed exposition of ...
Page 1629
Nelson Dunford, Jacob T. Schwartz. CHAPTER XIV Linear Partial Differential Equations and Operators 1. Introduction The Cauchy Problem , Local Dependence In this chapter , we shall discuss a variety of theorems having to do with linear ...
Nelson Dunford, Jacob T. Schwartz. CHAPTER XIV Linear Partial Differential Equations and Operators 1. Introduction The Cauchy Problem , Local Dependence In this chapter , we shall discuss a variety of theorems having to do with linear ...
Page 1639
... partial differential operator of the form T = Σ αγ ( π ) , Jsm then the formal partial differential operator T * = Σ ( −1 ) αл ( x ) J≤m is called the adjoint , or formal adjoint of τ . If r = 7 * , then 7 is said to be formally ...
... partial differential operator of the form T = Σ αγ ( π ) , Jsm then the formal partial differential operator T * = Σ ( −1 ) αл ( x ) J≤m is called the adjoint , or formal adjoint of τ . If r = 7 * , then 7 is said to be formally ...
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