Garcia, Stephan Ramon

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Claremont CollegesScholarship @ ClaremontPomona Faculty Publications and Research Pomona Faculty Scholarship2-17-2016Review: A C*-algebra approach to complexsymmetric operatorsStephan Ramon GarciaPomona CollegeThis Review is brought to you for free and open access by the Pomona Faculty Scholarship at Scholarship @ Claremont. It has been accepted forinclusion in Pomona Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, pleasecontact scholarship@cuc.claremont.edu.Recommended CitationMR3378818 Guo, K., Ji, Y., Zhu, S., A C*-algebra approach to complex symmetric operators, Trans. Amer. Math. Soc. 367 (2015), no.10, 6903-6942. (Reviewer: Stephan R. Garcia)Previous Up NextCitationsFrom References: 2From Reviews: 1MR3378818 (Review) 47C10 47A45 47A58 47B37Guo, Kunyu [Guo, Kun Yu] (PRC-FUDAN-SM) ;Ji, Youqing [Ji, You Qing] (PRC-JIL) ; Zhu, Sen (PRC-JIL)A C∗-algebra approach to complex symmetric operators. (English summary)Trans. Amer. Math. Soc. 367 (2015), no. 10, 6903–6942.This paper not only answers several open questions about complex symmetric operatorsbut also provides a number of new tools and ideas for their study. Let H denotea separable complex Hilbert space. A conjugate-linear map C:H → H is called aconjugation if C2 = I and 〈Cx,Cy〉= 〈y, x〉 for all x, y ∈ H. A bounded linear operatorT on H is a complex symmetric operator if T = CT ∗C.For a fixed H, let CSO denote the set of all complex symmetric operators on H; notethat T ∈ CSO if and only if there is a conjugation C on H so that T = CT ∗C. The normclosure problem for complex symmetric operators [S. R. Garcia and W. R. Wogen, J.Funct. Anal. 257 (2009), no. 4, 1251–1260; MR2535469] asked whether or not CSO isnorm closed in B(H); let CSO denote the closure of CSO with respect to the operatornorm. In [Proc. Amer. Math. Soc. 140 (2012), no. 5, 1705–1708; MR2869154], S. Zhu,C. G. Li and Y. Q. Ji demonstrated that the Kakutani shift, which is not complexsymmetric, belongs to CSO. Shortly thereafter, Garcia and Poore [Proc. Amer. Math.Soc. 141 (2013), no. 2, 549; MR2996959] obtained another example of an operator inCSO\CSO.A unilateral weighted shift T with nonzero weights {αk}∞k=1 is approximately Kakutaniif for each n≥ 1 and  > 0 there exists N ∈ N so that 0< |αn|<  and so that1≤ k ≤ n =⇒ ∣∣|αk| − |αN−k|∣∣< .This concept was introduced by Garcia and Poore in [J. Funct. Anal. 264 (2013), no. 3,691–712; MR3003733], where they conjectured that every irreducible weighted shiftin CSO is approximately Kakutani. Among many other things, this paper providesa positive resolution to this conjecture. To do this, the authors introduce a variety ofC∗-algebraic techniques to the study of complex symmetric operators and their relatives.A crucial observation is that if T ∈ CSO, then in many important cases T is actuallya small compact perturbation of an operator in CSO. Given a subset E of B(H), let Ecdenote the compact closure of E . This is the set of all A ∈ B(H) for which given any  >0 there is a compact K with ‖K‖<  and A+K ∈ E . The authors consider the questionof whether CSO and CSOccoincide and they provide positive answers for several specialclasses of operators.We say that A ∈ B(H) is called a transpose of T if A= CT ∗C for some conjugation Con H. Representing A and T with respect to an orthonormal basis of H, each of whoseelements is fixed by C, confirms that the operator transpose is a simple generalizationof the corresponding concept for matrices. We say that T is UET if T is unitarilyequivalent to T t (any two transposes are unitarily equivalent, so there is no ambiguityhere). This concept was introduced by Garcia and J. E. Tener [J. Operator Theory68 (2012), no. 1, 179–203; MR2966041]. One result of the authors here is a canonicaldecomposition of essentially normal operators that are UET. As a natural extension,we say that T is AUET if T is approximately unitarily equivalent to T t. The authorsprovide the following important connection: if C∗(T ) contains no nonzero compactoperators, then T ∈ CSO if and only if T is AUET.An operator T is g-normal if it satisfies ‖p(T ∗, T )‖= ‖p˜(T, T ∗)‖ for any polynomial pin two free variables, in which p˜(x, y) is obtained from p(x, y) by conjugating each coef-ficient. If T is a complex symmetric operator, then T is g-normal [S. R. Garcia, B. Lutzand D. Timotin, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1749–1756; MR3168480].The norm limit of g-normal operators is g-normal, so each operator in CSO is g-normal.Is every g-normal operator on H in CSO? The authors show that this is not the case,although the situation is more interesting than it seems. For instance, if C∗(T ) does notcontain a nonzero compact operator, then T ∈ CSO if and only if T is g-normal.The authors prove thatCSO⊂ CSO∪UET⊆AUET⊂ g-normal operatorsand that each inclusion is proper. The main results of the paper concern the relationsamong these classes and they provide many illuminating examples. For certain classesof operators (e.g., weighted shifts, essentially normal operators) the authors go muchfurther and obtain decompositions for these operators. There is much more in thispaper than can be described in a brief summary. It is certainly bound to be one ofthe more influential articles on the subject of complex symmetric operators and theirrelatives. Stephan R. GarciaReferences1. William Arveson, An invitation to C∗-algebras, Springer-Verlag, New York, 1976.Graduate Texts in Mathematics, No. 39. MR0512360 (58 #23621) MR05123602. Jeffrey L. Boersema, The range of united K-theory, J. Funct. Anal. 235 (2006), no.2, 701–718, DOI 10.1016/j.jfa.2005.12.012. 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Review: A C*-algebra approach to complex symmetric operators

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Review: A C*-algebra approach to complex symmetric operators

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