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Let $(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ be two complex Banach spaces such that $X\hookrightarrow Y$ and $X$ is dense in $Y$. Let $T:Y\to Y$ be a bounded linear operator that leaves $X$ invariant, i.e. $T(X)\subset X$. Furthermore, suppose that the restriction $T|_X$ is a bounded operator on $X$. Is it possible to say something about the relation of the spectral radii $r_Y(T)$ and $r_X(T|_X)$? In particular, is it possible that $r_X(T|_X)>r_Y(T)$?

Remark: If one restricts the operator to a subspace the point spectrum can only decrease but it's not clear whether this holds for the whole spectrum.

Daniele Tampieri
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Jake28
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1 Answers1

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Yes. Let $Y = \ell^1$, let $X$ be those elements such that ${\|x\|}_X = \sum_{n=1}^\infty 2^n |x_n|$ is finite, and let $T$ be the right shift: $$ T((x_1,x_2,\dots)) = (0,x_1,x_2,\dots) .$$ Then ${\|Tx\|}_Y = {\|x\|}_Y$ and ${\|Tx\|}_X = 2 {\|x\|}_X$. Hence $r_X(T|_X) = \lim_{k\to \infty} {\|T^k\|}_{X\to X}^{1/k} = 2 $, whereas $r_Y(T) = 1$.

Stephen Montgomery-Smith
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