Positive Operators

Suppose $T \in \mathcal{L}(V)$. Prove that if both $T$ and $-T$ are positive operators, then $T = 0$.

Suppose $T \in \mathcal{L}\left(\mathbf{F}^{4}\right)$ is the operator whose matrix (with respect to the standard basis) is

Show that $T$ is an invertible positive operator.

Suppose $n$ is a positive integer and $T \in \mathcal{L}\left(\mathbf{F}^{n}\right)$ is the operator whose matrix (with respect to the standard basis) consists of all 1's. Show that $T$ is a positive operator.

Suppose $n$ is an integer with $n > 1$. Show that there exists an $n$-by-$n$ matrix $A$ such that all of the entries of $A$ are positive numbers and $A = A^{*}$, but the operator on $\mathbf{F}^{n}$ whose matrix (with respect to the standard basis) equals $A$ is not a positive operator.

Suppose $T \in \mathcal{L}(V)$ is self-adjoint. Prove that $T$ is a positive operator if and only if for every orthonormal basis $e_{1}, \ldots , e_{n}$ of $V$, all entries on the diagonal of $\mathcal{M}(T, (e_{1}, \ldots , e_{n}))$ are nonnegative numbers.

Prove that the sum of two positive operators on $V$ is a positive operator.

Suppose $S \in \mathcal{L}(V)$ is an invertible positive operator and $T \in \mathcal{L}(V)$ is a positive operator. Prove that $S+T$ is invertible.

Suppose $T \in \mathcal{L}(V)$. Prove that $T$ is a positive operator if and only if the pseudoinverse $T^{\dagger }$ is a positive operator.

Suppose $T \in \mathcal{L}(V)$ is a positive operator and $S \in \mathcal{L}(W, V)$. Prove that $S^{*} T S$ is a positive operator on $W$.

Suppose $T$ is a positive operator on $V$. Suppose $v, w \in V$ are such that

Prove that $v = w$.

Suppose $T$ is a positive operator on $V$ and $U$ is a subspace of $V$ invariant under $T$. Prove that $T|_{U} \in \mathcal{L}(U)$ is a positive operator on $U$.

Suppose $T \in \mathcal{L}(V)$ is a positive operator. Prove that $T^{k}$ is a positive operator for every positive integer $k$.

Suppose $T \in \mathcal{L}(V)$ is self-adjoint and $\alpha \in \mathbf{R}$.

Suppose $T$ is a positive operator on $V$ and $v_{1}, \ldots , v_{m} \in V$. Prove that

Suppose $T \in \mathcal{L}(V)$ is self-adjoint. Prove that there exist positive operators $A, B \in \mathcal{L}(V)$ such that

Suppose $T$ is a positive operator on $V$. Prove that

Suppose that $T \in \mathcal{L}(V)$ is a positive operator. Prove that there exists a polynomial $p$ with real coefficients such that $\sqrt{T} = p(T)$.

Suppose $S$ and $T$ are positive operators on $V$. Prove that $S T$ is a positive operator if and only if $S$ and $T$ commute.

Show that the identity operator on $\mathbf{F}^{2}$ has infinitely many self-adjoint square roots.

Suppose $T \in \mathcal{L}(V)$ and $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $V$. Prove that $T$ is a positive operator if and only if there exist $v_{1}, \ldots , v_{n} \in V$ such that

for all $j, k = 1, \ldots , n$.

Suppose $n$ is a positive integer. The $n$-by-$n$ Hilbert matrix is the $n$-by-$n$ matrix whose entry in row $j$, column $k$ is $\frac{1}{j+k-1}$. Suppose $T \in \mathcal{L}(V)$ is an operator whose matrix with respect to some orthonormal basis of $V$ is the $n$-by-$n$ Hilbert matrix. Prove that $T$ is a positive invertible operator.

Suppose $T \in \mathcal{L}(V)$ is a positive operator and $u \in V$ is such that $\left\| u\right\| = 1$ and $\left\| T u\right\| \geq \left\| T v\right\|$ for all $v \in V$ with $\left\| v\right\| = 1$. Show that $u$ is an eigenvector of $T$ corresponding to the largest eigenvalue of $T$.

For $T \in \mathcal{L}(V)$ and $u, v \in V$, define $\left\langle u, v \right\rangle_{T}$ by $\left\langle u, v \right\rangle_{T} = \left\langle T u, v \right\rangle$.

Suppose $S$ and $T$ are positive operators on $V$. Prove that

Let $T$ be the second derivative operator in Exercise 31(b) in Section 7A. Show that $-T$ is a positive operator.

Exercise 7A.31: fix a positive integer $n$. In the inner product space of continuous real-valued functions on $[-\pi , \pi ]$ with inner product $\left\langle f, g \right\rangle = \int_{-\pi }^{\pi } f g$, let $V = \operatorname {span}(1, \cos x, \cos 2 x, \ldots , \cos n x, \sin x, \sin 2 x, \ldots , \sin n x)$. Part (b) defines $T \in \mathcal{L}(V)$ by $T f = f''$.