= Positive operator
{title2=$T\geq0$}
{wiki}
= Positive semidefinite operator
{synonym}
A bounded <Hermitian operator> $T$ on a <Hilbert space> is positive, also called a <positive semidefinite operator>, when
$$
q_T(x)=\langle Tx,x\rangle\geq0\qquad\text{for every }x\in H.
$$
The value must be real as well as nonnegative. This permits zero values on nonzero <vectors>: the zero <linear operator> is positive. For a bounded <Hermitian operator>, positivity is equivalent to its <spectrum> being contained in $[0,\infty)$; in finite <dimension>, this is the condition that all <eigenvalues> are nonnegative.https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/345c4a5141559a4069aeb3105acd8ef3_MIT18_102s21_lec21.pdf[Rodriguez, notes by Andrew Lin, MIT 18.102, lecture 21, Corollary 225, 2021]{ref}
The <Hermitian operator> diagram separates three cases. $D=\operatorname{diag}(1,-1)$ is <self-adjoint> but not positive, because $q_D((0,1)^{\mathrm T})=-1$. The <orthogonal projection> $P=\operatorname{diag}(1,0)$ is positive semidefinite, with $q_P(x)=|x_1|^2$, but is not strictly positive on every nonzero <vector>. The complex <Hermitian matrix> $H=\begin{pmatrix}2&i\\-i&2\end{pmatrix}$ is positive definite because $q_H(ae_1+be_3)=|a|^2+3|b|^2>0$ whenever $(a,b)\ne(0,0)$. For real <symmetric matrices>, MIT's lecture 27 relates these tests to the geometry of <quadratic forms> and their minima.https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/d163e012754258d3b374548504d8a18a_MIT18_06SCF11_Ses3.3sum.pdf[Strang, MIT 18.06SC, lecture 27 summary, Positive definite matrices and minima, pp. 1–2, 2011]{ref}
A fundamental source of <positive operators> is $T^*T$ for a <bounded linear operator> $T:H\to K$:
$$
(T^*T)^*=T^*T,\qquad \langle T^*Tx,x\rangle=\langle Tx,Tx\rangle=\|Tx\|^2\geq0.
$$
For the complex <matrix> $T=\begin{pmatrix}1&i\\2&1-i\end{pmatrix}$ used above,
$$
T^*T=\begin{pmatrix}5&2-i\\2+i&3\end{pmatrix},\qquad q_{T^*T}(x)=|x_1+ix_2|^2+|2x_1+(1-i)x_2|^2.
$$
Its <eigenvalues> are $4-\sqrt6$ and $4+\sqrt6$, both positive. Positivity organizes the squared error minimized by a <least-squares solution>: the same $T^*T$ occurs in the <normal equation for a linear inverse problem>. The <positive square root of an operator> gives the converse squared-length description, $q_T(x)=\|T^{1/2}x\|^2$ for every bounded <positive operator> $T$.
On a real <Hilbert space>, nonnegativity of $\langle Tx,x\rangle$ alone does not imply <self-adjointness>, so the <Hermitian operator> hypothesis is part of the definition. On a complex <Hilbert space>, real-valuedness of all these expressions already forces <self-adjointness> by the <polarization identity>.
Strict positivity and a uniform positive lower bound also differ in infinite <dimension>. On the <l2 sequence space>, the bounded <Hermitian operator> $(Tx)_n=x_n/n$ satisfies
$$
q_T(x)=\sum_{n=1}^{\infty}\frac{|x_n|^2}{n}>0\quad(x\ne0),\qquad q_T(e_n)=\frac1n\longrightarrow0.
$$
Consequently no $\gamma>0$ satisfies $q_T(x)\geq\gamma\|x\|^2$ for every $x$. Compare the <positive definite symmetric operator> and <uniformly positive definite symmetric operator> conditions.
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