If $0\leq A\leq B$ on Hilbert space and $A^{-1}$ exists, show that $A^{-1}\geq B^{-1}$
3038
For positive invertible operators $C\leq T$ on a Hilbert space, does it follow that $T^{-1}\leq C^{-1}$?
100257
Converse to Hilbert basis theorem
126032
Proving a certain inequality
98855
Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$
130327
Prove that $a^2 > b^2$ ; if $a > b \geq 0$.
81618
Prove $||a|-|b| | \leq |a-b|$
154327
$A^2X = X$ show that $AX = X$, if $A_{ij} > 0$ and $x_{i} > 0$
81818
Prove or disprove : $\det(A^k + B^k) \geq 0$
81775
Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$
17159
Proof of Closest Point Theorem in Hilbert Space
38850
How to show Von Neumann Trace inequality $ \text{Tr}(AB) \leq \sum_{i=1}^n \sigma_{A,i}\sigma_{B,i} $?
125707
A simple proof (hopefully) for $2^x \neq 0$
54380
Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$
33903
Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
73696
Given $(a_n)$ with $a_n \geq a_{n+1} \geq 0$ show that if $\sum \sqrt{a_n a_{n+1}} < \infty \implies \sum a_n < \infty$
44403
Prove that $|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$
72470
Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$