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

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Page 929

Invariant

Invariant

**subspaces**. If T is an operator in a B - space X , and if M is a closed linear**subspace**which is neither { 0 } nor X for which we have a TM C M , then M is called a ( non - trivial ) invariant**subspace**of X with respect to T.Page 930

this is far from clear , and it is of considerable interest to find non - trivial invariant

this is far from clear , and it is of considerable interest to find non - trivial invariant

**subspaces**for a given operator . ... Aronszajn and Smith [ 1 ] have shown that every compact operator has an invariant**subspace**even when o ( T ) ...Page 1228

There is a one - to - one correspondence between closed symmetric

There is a one - to - one correspondence between closed symmetric

**subspaces**S of the Hilbert space D ( T * ) which contain D ... Conversely , if S is a closed symmetric**subspace**of D ( T * ) including D ( T ) , put SI = S O ( D. D_ ) .### What people are saying - Write a review

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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 Ly(R 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 unique unit unitary vanishes vector zero