We know that for $T\in B(H)$. If $T$ is positive, then $T$ is self-adjoint and $\sigma(T)\subset R^{+}$. Do we have the inverse ie: if $T$ is self-adjoint and $\sigma(T)\subset R^{+}$. $T$ is positive.

where $T$ is said to be positive in case $\langle Tx,x\rangle\geq 0$ for all $x\in H$, $H$ is a Hilbert space.

Thank you very much for your help.

  • $\begingroup$ Do you know about the continuous functional calculus for a normal element of a C*-algebra? One can use this to immediately get that a normal element (in any C*-algebra) is positive if and only if it has non-negative spectrum. $\endgroup$
    – PStheman
    Sep 14, 2020 at 17:03
  • $\begingroup$ yes, I know a bit of the functional calculus. $\endgroup$ Sep 14, 2020 at 17:35

1 Answer 1


Theorem: Let $T$ be a bounded self-adjoint linear operator on a Hilbert space. Then $$ \lambda=\inf_{\|x\|=1}\langle Tx,x\rangle $$ is in the spectrum of $T$. Therefore, if $\sigma(T)\subset[0,\infty)$, then $\inf_{\|x\|=1}\langle Tx,x\rangle \ge 0$.

Proof: To prove what you want, let $\lambda=\inf_{\|x\|=1}\langle Tx,x\rangle$, and note that $\lambda$ satisfies $$ 0 \le \langle (T-\lambda I)x,x\rangle. $$ Therefore $[x,y] = \langle (T-\lambda I)x,x\rangle$ has all the properties of an inner product, except possibly strict positivity ($[x,x] \ge 0$ is always non-negative.) So the Cauchy-Schwarz inequality holds: $$ |[x,y]|^2 \le [x,x][y,y] \\ |\langle (T-\lambda I)x,y\rangle|^2 \le \langle (T-\lambda I)x,x\rangle\langle(T-\lambda I)y,y\rangle $$ Now set $y=(T-\lambda I)x$ in order to obtain \begin{align} \|(T-\lambda I)x\|^4 &\le \langle (T-\lambda I)x,x\rangle\cdot\langle(T-\lambda I)^2x,(T-\lambda I)x\rangle \\ &\le \langle(T-\lambda I)x,x\rangle\|(T-\lambda I)\|\|(T-\lambda I)x\|\|(T-\lambda I)x\| \\ \|(T-\lambda I)x\|^2 &\le\|(T-\lambda I)\|\langle(T-\lambda I)x,x\rangle \end{align} If you choose a sequence of unit vectors $\{ x_n\}$ so that $\langle (T-\lambda I)x_n,x_n\rangle\rightarrow 0$, it follows that $(T-\lambda I)x_n\rightarrow 0$, which forces $\lambda\in\sigma(T)$. $\;\;\blacksquare$

  • $\begingroup$ thank you very much, could you give me title of the book where i can find this and more to improve my knowledge.? $\endgroup$ Sep 14, 2020 at 18:04
  • $\begingroup$ @TryingToLearn : What is your background? Taylor and Lay, Functional Analysis is probably out of print. It's a classical introduction for many topics in Functional Analysis. Based on the fact that Amazon has a used copy listed for $855, I assume it is out of print, and posting this download link would be just fine. bookfi.net/dl/1463459/c0a8f7 $\endgroup$ Sep 14, 2020 at 19:13
  • $\begingroup$ thank you very much, I started reading Walter Rudin and Conway. I will look with attention at Taylor and Lay, Functional Analysis. If you have any other suggestions related to functional analysis and operator theory, it will be great. $\endgroup$ Sep 14, 2020 at 20:02
  • $\begingroup$ @TryingToLearn : I would also highly recommend the text by Peter Lax. He has a way of making things seem natural. $\endgroup$ Sep 14, 2020 at 20:14
  • $\begingroup$ @TryingToLearn : If you are interested in an excellent introduction to Spectral Theory that is aimed toward Operator Algebras, then you can't go wrong with William B. Arveson's "A Short Course in Spectral Theory". Arveson was an amazing talent in Operator Algebras, and he had a way of making things look simple in a field that is not simple. $\endgroup$ Sep 14, 2020 at 20:44

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