A positive operator is a linear operator $T$ on a Hilbert space such that for all vectors $x$ in the space, the inner product $ig\langle Tx, x \big\rangle$ is non-negative. This concept is crucial because it leads to various significant results in functional analysis, including insights into the spectrum of operators, their polar decompositions, and the spectral theorem, all of which help us understand the structure and behavior of bounded self-adjoint operators.
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