طيف المتباينات المتساوية الصارمة من الرتبة m على فضاءات هلبرت ذات البُعد المنتهي

Abstract

Abstract

In this paper, we investigate the spectral properties of strict m-isometries acting on finite dimensional complex Hilbert spaces. An operator T on a Hilbert space H is called an m-isometry if it satisfies a polynomial identity of the form β_m(T) = 0 involving alternating sums of the powers of the product T*T. When T satisfies this identity for some positive integer m but fails to satisfy it for m−1, T is called a strict m-isometry.

We provide a comprehensive characterization of the spectrum σ(T), the point spectrum σ_p(T), the approximate point spectrum σ_ap(T), and the residual spectrum σ_r(T) of such operators. We prove that for a strict m-isometry on a finite dimensional Hilbert space, every eigenvalue lies on the unit circle, and we establish precise conditions under which the three spectral components coincide. We further derive sharp bounds on the size of Jordan blocks in the Jordan normal form of T and show that these bounds are achieved by specific families of operators.

Additionally, we study the operator norms of powers of strict m-isometries, establish polynomial growth estimates, and examine the interplay between the order m of the isometry, the Jordan structure, and the geometry of the associated eigenspaces. Applications to discrete-time dynamical systems, stability theory, and Hilbert space geometry are discussed. Our results extend and unify several known theorems in the literature.