Gradient flows for bounded linear evolution equations (Q2107779)

A great amount of literature is dedicated to proving that certain evolution equations can be written as a gradient flow; see for example [\textit{T. Bárta} et al., Monatsh. Math. 166, No. 1, 57--72 (2012; Zbl 1253.37019); \textit{C. Cancès} et al., Arch. Ration. Mech. Anal. 233, No. 2, 837--866 (2019; Zbl 1459.76171); \textit{R. Jordan} et al., SIAM J. Math. Anal. 29, No. 1, 1--17 (1998; Zbl 0915.35120); \textit{J. Maas}, J. Funct. Anal. 261, No. 8, 2250--2292 (2011; Zbl 1237.60058); \textit{A. Mielke}, Calc. Var. Partial Differ. Equ. 48, No. 1--2, 1--31 (2013; Zbl 1282.60072)]. In [\textit{T. Bárta} et al., Monatsh. Math. 166, No. 1, 57--72 (2012; Zbl 1253.37019)], it is proven that every ordinary differential equation on a finite-dimensional manifold is a gradient flow if it possesses a strict Lyapunov function. Here, in paper, we answer the question of which linear equations of the form \(\dot{x}(t))=Ax(t)\) in a possibly infinite-dimensional separable Hilbert space H can be written as a gradient flow if the evolution operator \(A:H\to H\) is linear and bounded. For a infinite-dimensional separable Hilbert space H in Zbl. 1078.60003, the concepts of a gradient system, a canonical gradient system, a generalized gradient system were introduced. In Theorem 1, the authors investigated cases where these concepts coincide and obtained different representations of the operator In Theorem 2, one of the cases of Theorem 1 is clarified.