Bulletin of the Iranian Mathematical Society
Year:2006 | Volume:32 | Issue:2|Papers Count:6

In 1989, T. J. Ransford presented a short proof of Johnson's uniqueness of norm theorem. We follow the same method to show that if A and B are Fréchet algebras, B is semisimple and T: A → B is a surjective homomorphism with a certain condition, then T is continuous. In particular, when A is a Banach algebra we conclude that every epimorphism T: A → B is automatically continuous and hence every semisimple Banach algebra has a unique topology as a Fréchet algebra, which is an extension of Johnson's uniqueness of norm theorem.If A and B are Banach algebras, B is semisimple and T: A → B is a dense range homomorphism, then the continuity of T is a long-standing open question. In this work we give a positive answer to this open question with an extra condition on B and then present a partial answer to the well-known Michael's problem. We also obtain similar results for dense range homomorphisms of Fréchet algebras. Finally, we show that if the above question on the continuity of dense range homomorphisms of Banach algebras has a positive answer then the same question has a positive answer for Fréchet algebras.

Recently, Dhage [1] introduced the concept of D-metric. Afterwards, many authors [4, 5, 6] proved some fixed point theorems in these spaces. In this paper, using the concept of D-metric, we define a Δ–distance on a complete D-metric space which is a generalization of the concept of ω–distance due to Kada, Suzuki and Takahashi [2]. This generalization is non trivial because a D-metric does not always define a topology, and even when it does, this topology is not necessarily Hausdorff (see [3] and [4, Ch.1]). Using the concept of Δ-distance, we prove a fixed point theorem, which is the main result of this paper.We state the definition of D–metric, Δ–distance and prove a lemma. For more information on D–metrics, we refer the reader to [1, 4, 6].

For X, Y Î Mnm (:= Mnm(R)), we say X is weakly matrix majorized or matrix majorized from the left by Y and write X <e Y , if X = RY for some row stochastic matrix R. Also we write X ~e Y if X <e Y <e X. A mapping T: Mnm → Mnm is said to be a strong preserver of <e, if {X Î Mnm: X <e A} = {X Î Mnm: TX <e TA} for all A Î Mnm. Two such strong preservers T and Τ are called equivalent if TX ~e Τ X for all X Î Mnm. It is shown that if m≥ 2 and if T: Mnm → Mnm is a surjective (not necessarily linear) strong preserver of <e, then T −T0 is equivalent to a linear strong preserver of <e.

Let Mn,m be the set of all n×m matrices with entries in F, where F is the field of real or complex numbers. In this paper we introduce the relation gs-majorization onMn,m. We study some properties of this relation on Mn,m. We also characterize all linear operators that preserve (or strongly preserve) gs-majorization on Mn,m.

We improve Park and Bae's common fixed point theorem which is a generalization of Meir and Keeler's fixed point theorem. We extend Kannan's fixed point theorem to a common fixed point theorem of two commuting maps. Also, using the notion of biased mappings, we prove another common fixed point theorem.  

For weighted composition operator uCj: f « u.(f o j) on Lp(S), we give a necessary and sufficient condition to have the Hyers-Ulam stability.