On the boundedness of the local resolvent function

For a hyponormal operatorT on a complex Hilbert spaceH, we show that if the spectrum ofT has empty interior, then the local resolvent function,\(\hat x_T \), is unbounded for everyx∈H{0}. In particular, ifT is selfadjoint, then\(\hat x_T \) is unbounded for every nonzerox. The converse implication holds for a normal operator, but it is not true in general. Moreover, we give an example of an operatorT inc0 whose spectrum has empty interior, but there exists a nonzero vector,x, so that\(\hat x_T \) is bounded.