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A positive answer on Nirenberg’s problem on expansive mappings in Hilbert spaces. (English) Zbl 1523.47032

In the book [Topics in nonlinear functional analysis. New York: Courant Institute of Mathematical Sciences, New York University (1974; Zbl 0286.47037)], L. Nirenberg formulated the following problem that remained open for almost 50 years: Let \(X\) be a separable Hilbert space, \(T : X \to X\) be a continuous expanding map, i.e., \(\|Tx - Ty\| \ge \|x - y\|\) for all \(x, y \in X\). Is it true that, if \(T(X)\) has non-empty interior, then \(T\) must be surjective?
The problem was solved in the negative in [D. Ives and D. Preiss, Proc. Am. Math. Soc. 149, No. 1, 301–310 (2021; Zbl 1471.46069)] by means of a counterexample. In that example, the interior of \(T(X)\) is a bounded set.
In the paper under review, the author claims to prove that, under the additional condition of unboundedness of the interior of \(T(X)\), the answer to Nirenberg’s problem is positive.
Unfortunately, the announced statement comes into contradiction with Ives and Preiss’ example for the following reason. Consider the Hilbert space \(H\) which is the Cartesian product of the space \(X\) from Ives and Preiss’ example with some other Hilbert space \(Y\), that is, \( H = X \times Y\) equipped with the standard norm \(\| (x,y)\| = \sqrt{\| x \|^2 + \| y \|^2}\). Define the mapping \(T_1 : H \to H\) by the formula \[ T_1 (x,y) = (Tx, y), \] where \(T : X \to X\) is the continuous expanding non-surjective map with non-empty interior of the range constructed by Ives and Preiss. Then \(T_1\) is continuous and expanding. \(T_1\) is not surjective because \(T_1( H) = T(X) \times Y\). Finally, the interior of \(T_1(H)\) is equal to the Cartesian product of the interior of \(T(X)\) and the space \(Y\): \[ \mathring{ T_1(H)} = \mathring{T(X)} \times Y, \] which is an unbounded set.

MSC:

47B02 Operators on Hilbert spaces (general)
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

References:

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