Positive operators Sample Clauses

Positive operators. An important sub-space of B(H) is the space of selfadjoint operators. Their spectral theory tells us that the spectrum of a self- adjoint operator is a subset of the real line. For this reason, we define a partial order, similar to the one of real numbers. This is done as follows:‌ Definition 1.2.18. Let A ∈ B(H). Then A is said to be non-negative, A ≥ 0, if and only if (Af, f)H ≥ 0, for any f ∈ H. Similarly, we say A is positive, A > 0, if and only if (Af, f) > 0. Moreover, if A, B ∈ B(H) are selfadjoint, then we write A ≤ B (resp. A < B) if and only if B − A ≥ 0 (resp. B − A > 0).
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Related to Positive operators

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