Previous |  Up |  Next

Article

Keywords:
elementary operators; ultraweak closure; weak closure; quasi-adjoint operator
Summary:
Let $L(H)$ denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space $H$ into itself. Given $A\in L(H)$, we define the elementary operator $\Delta _A\colon L(H)\longrightarrow L(H)$ by $\Delta _A(X)=AXA-X$. In this paper we study the class of operators $A\in L(H)$ which have the following property: $ATA=T$ implies $AT^{\ast }A=T^{\ast }$ for all trace class operators $T\in C_1(H)$. Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of $\Delta _A$ is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints operators.
References:
[1] Anderson, J. H., Bunce, J. W., Deddens, J. A., Williams, J. P.: $C^{\ast}$-algebras and derivation ranges. Acta. Sci. Math. (Szeged) 40 (1978), 211-227. MR 0515202 | Zbl 0406.46048
[2] Apostol, C., Fialkow, L.: Structural properties of elementary operators. Canad. J. Math. 38 (1986), 1485-1524. DOI 10.4153/CJM-1986-072-6 | MR 0873420 | Zbl 0627.47015
[3] Berens, H., Finzel, M.: A problem in linear matrix approximation. Math. Nachr. 175 (1995), 33-46. DOI 10.1002/mana.19951750104 | MR 1355011 | Zbl 0838.47015
[4] Bouali, S., Bouhafsi, Y.: On the range of the elementary operator $X\mapsto AXA-X$. Math. Proc. Roy. Irish Acad. 108 (2008), 1-6. MR 2372836 | Zbl 1189.47033
[5] Dixmier, J.: Les $C^{\ast}$-algèbres et leurs représentations. Gauthier Villars, Paris (1964). MR 0171173 | Zbl 0152.32902
[6] Douglas, R. G.: On the operator equation $S^{\ast}XT=X$ and related topics. Acta. Sci. Math. (Szeged) 30 (1969), 19-32. MR 0250106 | Zbl 0177.19204
[7] Duggal, B. P.: On intertwining operators. Monatsh. Math. 106 (1988), 139-148. DOI 10.1007/BF01298834 | MR 0968331 | Zbl 0652.47019
[8] Duggal, B. P.: A remark on generalised Putnam-Fuglede theorems. Proc. Amer. Math. Soc. 129 (2001), 83-87. DOI 10.1090/S0002-9939-00-05920-7 | MR 1784016 | Zbl 0958.47015
[9] Embry, M. R., Rosenblum, M.: Spectra, tensor product, and linear operator equations. Pacific J. Math. 53 (1974), 95-107. DOI 10.2140/pjm.1974.53.95 | MR 0353023
[10] Fialkow, L.: Essential spectra of elementary operators. Trans. Amer. Math. Soc. 267 (1981), 157-174. DOI 10.1090/S0002-9947-1981-0621980-8 | MR 0621980 | Zbl 0475.47002
[11] Fialkow, A., Lobel, R.: Elementary mapping into ideals of operators. Illinois J. Math. 28 (1984), 555-578. MR 0761990
[12] Fialkow, L.: Elementary operators and applications. (Editor: Matin Mathieu), Procceding of the International Workshop, World Scientific (1992), 55-113. MR 1183937
[13] Fong, C. K., Sourour, A. R.: On the operator identity $\sum A_kXB_k=0$. Canad. J. Math. 31 (1979), 845-857. DOI 10.4153/CJM-1979-080-x | MR 0540912 | Zbl 0368.47024
[14] Genkai, Z.: On the operators $X\mapsto AX-XB$ and $ X\mapsto AXB-X$. Chinese J. Fudan Univ. Nat. Sci. 28 (1989), 148-154.
[15] Magajna, B.: The norm of a symmetric elementary operator. Proc. Amer. Math. Soc. 132 (2004), 1747-1754. DOI 10.1090/S0002-9939-03-07248-4 | MR 2051136 | Zbl 1055.47030
[16] Mathieu, M.: Rings of quotients of ultraprime Banach algebras with applications to elementary operators. Proc. Centre Math. Anal., Austral. Nat. Univ. Canberra 21 (1989), 297-317. MR 1022011 | Zbl 0701.46027
[17] Mathieu, M.: The norm problem for elementary operators. Recent progress in functional analysis (Valencia 2000) 363-368 North-Holland Math. Stud. 189, North-Holland, Amsterdam (2001). DOI 10.1016/S0304-0208(01)80061-X | MR 1861772 | Zbl 1011.47027
[18] Stachò, L. L., Zalar, B.: On the norm of Jordan elementary operators in standard operator algebra. Publ. Math. Debrecen 49 (1996), 127-134. MR 1416312
Partner of
EuDML logo