Norms of powers of self-adjoint operator

Question

Let $X$ be a Hilbert space, $T\in L(X)$ be a linear, continuous, self-adjoint operator. Due to $$ \|T^2\| = \|TT^*\| = \|T\|^2, $$ one can prove by induction $$ \|T^{2^n}\|=\|T\|^{2^n}. $$ This can be used to deduce $\lim_{n\to\infty}\|T^n\|^{1/n}=\|T\|$.

My question is: can one prove $$ \|T^n\|=\|T\|^n $$ for all $n$ in the non-compact case ($T$ not compact) by elementary calculations like above? That is, without invoking spectral theorems or decompositions.

Answer 1

We have that

$$\lim_{n\to\infty}\|T^n\|^{1/n}= \inf_{n \in \mathbb N}\|T^n\|^{1/n}.$$

Hence, if

$$\lim_{n\to\infty}\|T^n\|^{1/n}=||T||,$$ then, for each $n \in \mathbb N,$

$$||T|| \le \|T^n\|^{1/n},$$

or

$$||T||^n \le \|T^n\|.$$

Since $$||T^n|| \le ||T||^n,$$

we get

$$\|T^n\|=\|T\|^n.$$