Hilbert space Let X be a Hilbert space. Recall that T B(X) is positive (in symbo
ID: 1945345 • Letter: H
Question
Hilbert space Let X be a Hilbert space. Recall that T B(X) is positive (in symbols, T 0) iff (Tx, x) 0 for all x X. Prove: The positive operator T is non-singular (i.e., invertible in B(X)) iff T – I 0 for some > 0 (one writes also T I to express the last relation). Hilbert space Let X be a Hilbert space. Recall that T B(X) is positive (in symbols, T 0) iff (Tx, x) 0 for all x X. Prove: The positive operator T is non-singular (i.e., invertible in B(X)) iff T – I 0 for some > 0 (one writes also T I to express the last relation).Explanation / Answer
The operator T is non-singular iff 0 ? p(T). Since p(T) is open, this is equivalent to the existence of e > 0 such that the disc B(0, e) is contained in p(T). Since T is positive, s(T) ? [0,8), and therefore, in that case, T is non-singular iff s(T) ? [e,8), that is, iff s(T) – e ? [0,8), i.e., iff s(T – e I) ? [O, 8). Since T – e I is self-adjoint (because T is self-adjoint!), the latest condition is equivalent to the positivity of T – e I.
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