Positive Operator

Introduction

Let $T$ be a self-adjoint operator. Then, by the properties of self-adjoint operators, the following holds.

$$ \braket{Tx, x} \in \mathbb{R} $$

That is, we can speak about whether the above value is positive or negative. Based on this, let us define a positive operator as follows.

Definition

Let $H$ be a Hilbert space and let $T: H \to H$ be a self-adjoint operator. When $T$ satisfies the following, it is called a positive operator, and we write $T \ge 0$.

$$ \braket{Tx, x} \ge 0, \quad \forall x \in H $$

Explanation

If $H$ is finite-dimensional, then $T$ becomes a matrix and $x$ becomes a vector. Since the inner product of two vectors is $\braket{x, y} = x^{\ast}y$, the condition for being a positive operator is as follows.

$$ \braket{Tx, x} = (Tx)^{\ast} x = x^{\ast} T^{\ast} x = x^{\ast} T x \ge 0 $$

That is, a positive operator is a generalization of a positive definite matrix.

Properties

(a) By the linearity of the inner product, the sum of two positive operators is also a positive operator. $$ T \ge 0, \quad S \ge 0 \implies T + S \ge 0 $$