##
**Linear relations in the Calkin algebra for composition operators.**
*(English)*
Zbl 1115.47023

The purpose of this interesting paper is to consider the following problem: When is a finite linear combination of composition operators, acting on the Hardy space or on the standard weighted Bergman spaces on the unit disc, a compact operator? The set of composition operators, as a subset of the algebra of all bounded operators, has no obvious additive or linear structure. Nevertheless, J. Moorhouse [J. Funct. Anal. 219, No. 1, 70–92 (2005; Zbl 1087.47032)] observed additive structure in the weighted Bergman spaces, modulo the ideal of compact operators, and characterized the pairs of analytic self-maps \(\varphi\) and \(\psi\) on the unit disc such that the difference \(C_{\varphi} - C_{\psi}\) is compact.

In the present paper, analogous results for the Hardy space \(H^2\) are obtained, and the authors are able to pass from additive to linear structure in the Calkin algebra. The problem of compact difference of composition operators was raised by Shapiro and Sundberg and by MacCluer in 1989–90. The pseudo-hyperbolic distance between the values of \(\varphi\) and \(\psi\) played some role. These results are related to the characterizations of those \(\varphi\) and \(\psi\) such that the corresponding composition operators lie in the same connected component of the topological space of composition operators endowed with the operator norm. In order to obtain several results due to Moorhouse [loc. cit.] for \(H^2\), different methods are needed. An application of Clark measures is essential for the authors.

In Section 3, they obtain essential norm estimates for weighted composition operators on \(H^2\) similar to the Cima–Matheson essential norm formula for unweighted composition operators. The \(H^2\) results are presented in Section 4. Section 5 is devoted to the question of when a finite linear combination of composition operators is compact. The authors further develop ideas of MacCluer and obtain lower bounds of the essential norm in terms of first and higher order boundary data. A special class \(S\) of analytic self-maps on the unit disc, which have sufficient data at every point of the boundary at which the map makes significant contact with the boundary, is introduced. For pairs of analytic self-maps in the class \(S\), the obstructions to the essential norm of the difference being small are clarified completely, as well as conditions under which it must be small or 0.

The question mentioned at the beginning of this review is then completely solved for finite combinations of composition operators with symbols in the class \(S\). In the final Section 6, in the case when the symbol \(\varphi\) lies in a certain subclass of the class \(S\), the authors determine when another composition operator \(C_{\psi}\) belongs to the same component as \(C_{\varphi}\) both for Hardy and weighted Bergman spaces.

In the present paper, analogous results for the Hardy space \(H^2\) are obtained, and the authors are able to pass from additive to linear structure in the Calkin algebra. The problem of compact difference of composition operators was raised by Shapiro and Sundberg and by MacCluer in 1989–90. The pseudo-hyperbolic distance between the values of \(\varphi\) and \(\psi\) played some role. These results are related to the characterizations of those \(\varphi\) and \(\psi\) such that the corresponding composition operators lie in the same connected component of the topological space of composition operators endowed with the operator norm. In order to obtain several results due to Moorhouse [loc. cit.] for \(H^2\), different methods are needed. An application of Clark measures is essential for the authors.

In Section 3, they obtain essential norm estimates for weighted composition operators on \(H^2\) similar to the Cima–Matheson essential norm formula for unweighted composition operators. The \(H^2\) results are presented in Section 4. Section 5 is devoted to the question of when a finite linear combination of composition operators is compact. The authors further develop ideas of MacCluer and obtain lower bounds of the essential norm in terms of first and higher order boundary data. A special class \(S\) of analytic self-maps on the unit disc, which have sufficient data at every point of the boundary at which the map makes significant contact with the boundary, is introduced. For pairs of analytic self-maps in the class \(S\), the obstructions to the essential norm of the difference being small are clarified completely, as well as conditions under which it must be small or 0.

The question mentioned at the beginning of this review is then completely solved for finite combinations of composition operators with symbols in the class \(S\). In the final Section 6, in the case when the symbol \(\varphi\) lies in a certain subclass of the class \(S\), the authors determine when another composition operator \(C_{\psi}\) belongs to the same component as \(C_{\varphi}\) both for Hardy and weighted Bergman spaces.

Reviewer: José Bonet (Valencia)

### MSC:

47B33 | Linear composition operators |

47B32 | Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) |

30D55 | \(H^p\)-classes (MSC2000) |

47C05 | Linear operators in algebras |

### Keywords:

composition operators; Hardy space; weighted Bergman spaces; linear relations modulo the compact operators
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\textit{T. Kriete} and \textit{J. Moorhouse}, Trans. Am. Math. Soc. 359, No. 6, 2915--2944 (2007; Zbl 1115.47023)

### References:

[1] | A. B. Aleksandrov, Multiplicity of boundary values of inner functions, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 22 (1987), no. 5, 490 – 503, 515 (Russian, with English and Armenian summaries). |

[2] | Earl Berkson, Composition operators isolated in the uniform operator topology, Proc. Amer. Math. Soc. 81 (1981), no. 2, 230 – 232. · Zbl 0464.30027 |

[3] | Paul S. Bourdon, Components of linear-fractional composition operators, J. Math. Anal. Appl. 279 (2003), no. 1, 228 – 245. · Zbl 1043.47021 |

[4] | Paul S. Bourdon, David Levi, Sivaram K. Narayan, and Joel H. Shapiro, Which linear-fractional composition operators are essentially normal?, J. Math. Anal. Appl. 280 (2003), no. 1, 30 – 53. · Zbl 1024.47008 |

[5] | Joseph A. Cima and Alec L. Matheson, Essential norms of composition operators and Aleksandrov measures, Pacific J. Math. 179 (1997), no. 1, 59 – 64. · Zbl 0871.47027 |

[6] | Douglas N. Clark, One dimensional perturbations of restricted shifts, J. Analyse Math. 25 (1972), 169 – 191. · Zbl 0252.47010 |

[7] | Carl C. Cowen, Composition operators on \?², J. Operator Theory 9 (1983), no. 1, 77 – 106. · Zbl 0504.47032 |

[8] | Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. · Zbl 0873.47017 |

[9] | Peter L. Duren, Theory of \?^{\?} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. |

[10] | Pamela Gorkin and Raymond Mortini, Norms and essential norms of linear combinations of endomorphisms, Trans. Amer. Math. Soc. 358 (2006), no. 2, 553 – 571. · Zbl 1081.47029 |

[11] | Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. · Zbl 0040.16802 |

[12] | Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. · Zbl 0734.46033 |

[13] | Barbara D. MacCluer, Components in the space of composition operators, Integral Equations Operator Theory 12 (1989), no. 5, 725 – 738. · Zbl 0685.47027 |

[14] | Barbara MacCluer, Shûichi Ohno, and Ruhan Zhao, Topological structure of the space of composition operators on \?^{\infty }, Integral Equations Operator Theory 40 (2001), no. 4, 481 – 494. · Zbl 1062.47511 |

[15] | Barbara D. MacCluer and Joel H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), no. 4, 878 – 906. · Zbl 0608.30050 |

[16] | Jennifer Moorhouse, Compact differences of composition operators, J. Funct. Anal. 219 (2005), no. 1, 70 – 92. · Zbl 1087.47032 |

[17] | Jennifer Moorhouse and Carl Toews, Differences of composition operators, Trends in Banach spaces and operator theory (Memphis, TN, 2001) Contemp. Math., vol. 321, Amer. Math. Soc., Providence, RI, 2003, pp. 207 – 213. · Zbl 1052.47018 |

[18] | Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002. Model operators and systems; Translated from the French by Andreas Hartmann and revised by the author. · Zbl 1007.47001 |

[19] | Pekka J. Nieminen and Eero Saksman, On compactness of the difference of composition operators, J. Math. Anal. Appl. 298 (2004), no. 2, 501 – 522. · Zbl 1072.47021 |

[20] | A. G. Poltoratskiĭ, Boundary behavior of pseudocontinuable functions, Algebra i Analiz 5 (1993), no. 2, 189 – 210 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 2, 389 – 406. |

[21] | S. Saitoh, Integral transforms, reproducing kernels and their applications, Pitman Research Notes in Mathematics Series, vol. 369, Longman, Harlow, 1997. · Zbl 0891.44001 |

[22] | Donald Sarason, Composition operators as integral operators, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 545 – 565. · Zbl 0712.47026 |

[23] | Donald Sarason, Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10, John Wiley & Sons, Inc., New York, 1994. A Wiley-Interscience Publication. · Zbl 1253.30002 |

[24] | Jonathan E. Shapiro, Aleksandrov measures used in essential norm inequalities for composition operators, J. Operator Theory 40 (1998), no. 1, 133 – 146. · Zbl 0997.47023 |

[25] | Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375 – 404. · Zbl 0642.47027 |

[26] | Joel H. Shapiro and Carl Sundberg, Isolation amongst the composition operators, Pacific J. Math. 145 (1990), no. 1, 117 – 152. · Zbl 0732.30027 |

[27] | Joel H. Shapiro and Carl Sundberg, Compact composition operators on \?\textonesuperior , Proc. Amer. Math. Soc. 108 (1990), no. 2, 443 – 449. · Zbl 0704.47018 |

[28] | J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on \?², Indiana Univ. Math. J. 23 (1973/74), 471 – 496. · Zbl 0276.47037 |

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