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The zero product problem and the commuting problem for Toeplitz operators on the Bergman space over the unit disk are some of the most interesting unsolved problems. For bounded harmonic symbols these are solved but for general bounded symbols it is still far from being complete. This paper shows that the zero product problem holds for a special case where one of the symbols has certain polar decomposition and the other is a general bounded symbol. We also prove that the commutant of Tz+z is sum of powers of itself. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47L80.

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The study of the continuity of the farthest point mapping for uniquely remotal sets has been used extensively in the literature to prove the singletoness of such sets. In this article, we show that the farthest point mapping is not continuous even if the set is remotal, rather than being uniquely remotal. Consequently, we obtain some generalizations of results concerning the singletoness of remotal sets. In particular, it is proved that a compact set admitting a unique farthest point to its center is a singleton, generalizing the well known result of Klee.