Abstract
It is of historical interest that questions in semigroup perturbation theory led some years ago to a theory of antieigenvalues. That theory and those origins will be explained. Simultaneously we propose and pursue by similar considerations a new mechanism of irreversibility for quantum theory, based upon multiplicative disturbances.
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References
See, for instance, T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1980).
See, for instance, K. Yosida, Functional Analysis (Springer, Berlin, 1966).
K. Gustafson, Pacific J. Math. 24 (1968), 463.
A good accounting of the antieigenvalue theory may be found in the recent books: K. Gustafson, Lectures on Computational Fluid Dynamics, Mathematical Physics, and Linear Algebra (Kaigai, Tokyo, 1996)
K. Gustafson and D. Rao, Numerical Range: The Field of Values of Linear Operators and Matrices (Springer, Berlin, 1997).
The operator trigonometry and antieigenvalue theory lay mostly dormant from 1970 to 1990. Then the author found that the famous Kantorovich convergence rate for descent algorithms in numerical optimization schemes was trigonometric. See K. Gustafson, Proc. Fourth International Workshop in Analysis and Applications (eds.: C. Stanojevic and O. Hadzic), Novi Sad, Yugoslavia (1991), 57. This discovery has led to a fundamental new conceptualization of a wide class of iterative algorithms of wide use for solving large sparse systems Ax = b. See K. Gustafson, Num. Lin. Alg. with Applic. (1997), to appear.
By a general result of J.P. Williams, J. Math. Anal. Appl. 17 (1967), 214, σ(T −1 S) ⊂ W(S)/W(T) whenever 0 ∉ W(T). Here W(T) denotes the numerical range of an operator T. For selfadjoint T, W(T) is the convex hull of the spectrum σ(T). Thus in our situation, σ(BH) is real and positive.
E.P. Wigner, Canadian J. Math. 15 (1963), 313. The essential meaning of weakly positive operators T is best seen in the finite dimensional matrix case: σ(T) is real and positive and the eigenvectors of T form a complete set.
To be clear what we mean by this, we take the point of view of Nicolis and Prigogine, Exploring Complexity (Freeman, New York, 1989), 164, from which we quote “… the signature of irreversibility lies in the emergence of a dissipative semigroup description of an appropriately defined markovian process.” Here we only deal with the semigroup issue. For the corresponding markov processes
see I. Antoniou and K. Gustafson, Physica A 197, (1993), 153; Physica A (1997), to appear.
See E.B. Davies, One Parameter Semigroups (Academic Press, London, 1980), p. 155. By Davies’ Theorem, any Hamiltonian of the form iH — V with V bounded positive definite will generate a completely nonunitary contraction semigroup.
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© 1998 Springer-Verlag
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Gustafson, K. (1998). Semigroups and antieigenvalues. In: Bohm, A., Doebner, HD., Kielanowski, P. (eds) Irreversibility and Causality Semigroups and Rigged Hilbert Spaces. Lecture Notes in Physics, vol 504-504. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0106794
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DOI: https://doi.org/10.1007/BFb0106794
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