Abstract We study, for a continuous linear operator T acting on an F-space X , when the direct sum operator $$T\oplus T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>⊕</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> is recurrent on the direct sum space $$X\oplus X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊕</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> . In particular: we establish the analogous notion for recurrence to that of (topological) weak-mixing for transitivity/hypercyclicity, namely quasi-rigidity; and we construct a recurrent but not quasi-rigid operator on each separable infinite-dimensional Banach space, solving the $$T\oplus T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>⊕</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> -recurrence problem in the negative way.

Sophie Grivaux

Antoni López-Martínez

Alfred Peris

Analysis and Mathematical Physics

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The journal "Analysis and Mathematical Physics" is dedicated to publishing current research findings and selected high-quality survey articles in real, complex, and harmonic forms. The geometric analysis featured in this journal originates from or is applied to mathematical physics, aiming to foster in-depth dialogue among experts in these fields.

《Analysis and Mathematical Physics》期刊致力于以真实、复杂和谐波的形式发布当前的研究成果，以及精选的高质量调查文章。该期刊的几何分析内容源于或应用于数学物理，旨在促进这些领域专家之间的深入对话。

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数学	3
数学, 数学	3
数学, 应用数学	3

Questions in linear recurrence I: the $$T\oplus T$$-recurrence problem

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MATHEMATICS, APPLIED	2