• Bulletin of the Korean Mathematical Society
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• v.27 no.2
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• pp.151-155
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• 1990

Recently, H.Z.Ming [7] obtained a necessary and sufficient condition for the existence of a solution to a general operator equation. In the present paper, we obtain such conditions in general forms and give some examples. We begin with the well-known Fan-Browder fixed point theorem, from which we deduce two general theorems on such necessary and sufficient conditiions. We give some examples of such conditions, which are improved versions of fixed point theorems of Halpern-Bergman [5], Ky Fan [3], [4], Kaczynski [6], Reich [9], Schauder [10], Tychonoff [11], and Ming [7]. In fact, we restate Ming's result in its correct form. The following is known as the Fan-Browder fixed point theorem [1], [2].

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.77-95
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• 2012

For a bounded linear operator T we prove the following assertions: (a) If T is algebraically (p, k)-quasihyponormal, then T is a-isoloid, polaroid, reguloid and a-polaroid. (b) If $T^*$ is algebraically (p, k)-quasihyponormal, then a-Weyl's theorem holds for f(T) for every $f{\in}Hol({\sigma}T))$, where $Hol({\sigma}(T))$ is the space of all functions that analytic in an open neighborhoods of ${\sigma}(T)$ of T. (c) If $T^*$ is algebraically (p, k)-quasihyponormal, then generalized a-Weyl's theorem holds for f(T) for every $f{\in}Hol({\sigma}T))$. (d) If T is a (p, k)-quasihyponormal operator, then the spectral mapping theorem holds for semi-B-essential approximate point spectrum $\sigma_{SBF_+^-}(T)$, and for left Drazin spectrum ${\sigma}_{lD}(T)$ for every $f{\in}Hol({\sigma}T))$.