## Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |

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Page 972

From the preceding lemma Ke + it is seen that ( 7-1Xe + p ) ( x ) = pcx , [ x , p ] r - Xer Since characters have modulus

From the preceding lemma Ke + it is seen that ( 7-1Xe + p ) ( x ) = pcx , [ x , p ] r - Xer Since characters have modulus

**equal**to unity , it follows from Plancherel's theorem that { use + p ) } 2 ...Page 1017

( b ) The trace of an operator in En is

( b ) The trace of an operator in En is

**equal**to the sum of the numbers in the spectrum of the operator , if each number is counted according to its multiplicity as a root of the characteristic polynomial .Page 1454

I / T is a closed symmetric operator in Hilbert space , and T is bounded below , then a ( a ) the essential spectrum of T is a subset of the real axis which is bounded below ; ( b ) the deficiency indices of T are

I / T is a closed symmetric operator in Hilbert space , and T is bounded below , then a ( a ) the essential spectrum of T is a subset of the real axis which is bounded below ; ( b ) the deficiency indices of T are

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additive adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined eigenvalues element equal equation Exercise exists extension fact finite dimensional follows formal formal differential operator formula function function f given Hence Hilbert space Hilbert-Schmidt ideal identity independent inequality integral interval isometric isomorphism Lemma limit linear matrix measure multiplicity neighborhood norm normal operator obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solution spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero