Abstract. If an operator $T$ defined on the intersection of a suitable couple of Banach spaces $X$ and $Y$ has an extension $T_X \in{\cal L}(X)$ and an extension $T_Y \in{\cal L}(Y)$, there arises the question whether the spectrum $\sigma(T_X)$ of $T_X$ is equal to that of $T_Y$. This paper is concerned with this question. As a typical result, in the case where $X \cap Y$ is dense in $X$ and in $Y$, it is proved that if $T$ has an extension that is bounded from $X+Y$ into $X \cap Y$, then $\sigma(T_X)=\sigma(T_Y)$. This result has an application to the $L^p$-spectral independence of bounded operators. On the other hand, in the case where $Y$ is continuously and not necessarily densely embedded into $X$, it is proved that if $X$ and $T \in{\cal L}(X,Y)$ satisfy certain conditions, then $\sigma(T_X)=\sigma(T_Y)$. This result gives an example that shows an operator in ${\cal L}(L^\infty({\msbm R}^N))$ and its part in ${\cal L}(C_0({\msbm R}^N))$ have the same spectrum.

AMS Subject Classification (1991): 47A10, 47A25, 47G10

Keyword(s): spectrum of bounded operator, $L^p$-spectral independence, product of operators, integral operator

Received November 1, 2009, and in revised form May 6, 2010. (Registered under 6079/2009.)