Geoffrey R. Goodson

Transformations conjugate to their inverses have even essential values

Let T be an ergodic automorphism defined on a standard Borel probability space for which T and T−1 are isomorphic. We study the structure of the conjugating automorphisms and attempt to gain information about the structure of T . It was shown in Ergodic transformations conjugate to their inverses by involutions by Goodson et al. (Ergodic Theory and Dynamical Systems 16 (1996), 97–124) that if T is ergodic having simple spectrum and isomorphic to its inverse, and if S is a conjugation between T and T−1 (i.e. S satisfies TS = ST−1), then S2 = I, the identity automorphism. We give a new proof of this result which shows even more, namely that for such a conjugation S, the unitary operator induced by T on L2(X, μ) must have a multiplicity function whose essential values on the ortho-complement of the subspace {f ∈ L2(X, μ) : f(S2) = f} are always even. In particular, we see that S can be weakly mixing, so the corresponding T must have even maximal spectral multiplicity (regarding ∞ as an even number).

Inverse Conjugacies and Reversing Symmetry Groups

This paper arose from a course I gave on algebraic structures, where some of the results of Sections 1 and 2 and some examples from Section 4 were presented as exercises and then discussed in the classroom. In addition, the students were asked to calculate B(a) and C(a) for certain specific examples, sometimes with the aid of a software package. Generally B(a) is not a subgroup of G, and it may be empty. However, E(a) = B(a) U C(a) is a group, which is called the reversing symmetry group of a. In dynamical systems theory, the group element a represents the time evolution operator of a dynamical system. We present some results familiar to people working in time reversing dynamical systems, but our presentation is given in an

Quasi-real normal matrices and eigenvalue pairings

Abstract For square complex matrices A and B of the same size, commutativity-like relations such as AB=±BA, AB=±BA ∗ , AB=±BAT, AB=±BTA, etc., often cause a special structure of A to be reflected in some special structure for B. We study eigenvalue pairing theorems for B when A is quasi-real normal (QRN), a class of complex matrices that is a natural generalization of the real normal matrices. A new canonical form for QRN matrices is an essential tool in our development.

The Inverse-Similarity Problem for Real Orthogonal Matrices

INTRODUCTION. The aim of this article is to make accessible some recent results ([1], [2]) in the spectral theory of unitary operators. We do this by investigating the special case of real unitary matrices, i.e., real orthogonal matrices. Our development reinforces the importance of canonical forms and matrix decompositions in the undergraduate linear algebra curriculum. An orthogonal matrix Q (with entries from any ring) is a square matrix whose transpose is its inverse (QTQ = QQT = I). It is known that every square matrix U with entries from any field is similar to its transpose, that is, UA = AUT for some nonsingular matrix A; we say that such an A is a similarity between U and UT and we note that A may always be chosen to be symmetric (Theorem 1 of [6]; see 3.2.3 of [4] for the easier complex case). We are interested in the form such similarities A can have when U is real orthogonal, and what they tell us about the matrix U. Using the orthogonal canonical form of U, we show that A may be chosen to be a real orthogonal matrix satisfying A2 = I. Our main theorems (Theorems 1 and 2) give necessary and sufficient conditions for every real orthogonal similarity A between U and its transpose to be an involution (i.e., satisfy A2 = I). Finally, we investigate the case where there exist such similarities A that are not involutions. We show (Theorem 3) that the eigenvalues of U corresponding to eigenvectors of U in the orthogonal complement of the subspace {x: A2x = x} have even multiplicity. It may be that the results of this paper are well known, or follow easily from known results. Theorem 1, for example, follows from Theorem 2 of Taussky and Zassenhaus [6], which says: Every non-singular matrix transforming U into its transpose is symmetric if and only if the minimal polynomial of U is equal to its characteristic polynomial. A survey of results concerning the links between general matrices and their transposes is given in [5]. However, Theorem 3 may actually be new, and certainly the generalizations to the infinite dimensional situation, which we state in Section 2, are new (see [1] and [2]). Although the proofs of our theorems use matrix methods, their infinite dimensional analogs require the spectral theory of unitary operators. Throughout we shall be working with real orthogonal matrices belonging to the space of all n x n complex matrices Mnxn(C) Our vectors are in F= MnX1(C),

The converse of the inverse-conjugacy theorem for unitary operators and ergodic dynamical systems

We show that the converse to the main theorem of Ergodic transformations conjugate to their inverses by involutions, by Goodson et al. (Ergodic Theory and Dynamical Systems 16 (1996), 97-124), holds in the unitary category. Specifically it is shown that if U is a unitary operator defined on an L2 space which preserves real valued functions, and if U-S = SU implies S2 = I whenever S is another such operator, then U has simple spectrum. The corresponding result for measure preserving transformations is shown to be false. The counter-example we have involves Gaussian automorphisms. We show that a Gaussian automorphism is always conjugate to its inverse, so that the Inverse-Conjugacy Theorem is applicable to such maps having simple spectrum. Furthermore, there are Gaussian automorphisms having non-simple spectrum for which every conjugation of T with T-1 is an involution. ?0. INTRODUCTION Let U: H -* H be a unitary operator defined on a separable Hilbert space H. U is said to have simple spectrum if there exists an h E H for which Z(h) = H, where Z(h) is the closed linear span of the set {UTh: n E Z}. We are mainly interested in the case where the Hilbert space is a function space. Let (X, F, ,u) denote a standard Borel probability space, and let T: X -+ X be an invertible measure preserving transformation (automorphism). To say that the automorphism T has simple spectrum means that the unitary operator UT: L2(X, 1t) -* L2(X, ,u), UTf(x) = f(Tx), induced by T, has simple spectrum. It is known that if T has simple spectrum, then T is ergodic, and that unitary operators preserving real valued functions are conjugate to their inverses. The following theorem and corollary were given in [2] (see also [3] and [4] for related results). We use I to denote both the identity operator, and the identity automorphism. Inverse Conjugacy Theorem. Let U: L2(X, F, 1t) -+ L2(X, F, p) be a unitary operator having simple spectrum and preserving real valued functions. If S is another unitary operator preserving real valued functions and U1S SU, then S2 = J. Received by the editors October 28, 1997 and, in revised form, June 25, 1998. 1991 Mathematics Subject Classification. Primary 28D05; Secondary 47A35. (?2000 American Mathematical Society