Ideals of causal operators. (English) Zbl 0599.47067
The definitions and properties of various ideals of a nest algebra are reviewed. These correspond to strengthenings of the concept of causality. The interrelationships between these concepts are examined. An open question concerning these ideals is settled, namely that the strongly strictly causal operators form the largest of the classes commonly considered.
MSC:
| 47L30 | Abstract operator algebras on Hilbert spaces |
Keywords:
ideals of a nest algebra; strengthenings of the concept of causality; strongly strictly causal operatorsReferences:
| [1] | Deddens, J. A.,Another description of nest algebras. Lecture Notes in Mathematics No. 693, Springer-Verlag, New York, 1978, pp. 77–86. · Zbl 0405.47029 |
| [2] | Erdos, J. A.,Operators of finite rank in nest algebras. J. London Math. Soc.,43 (1968), 391–397. · Zbl 0169.17501 · doi:10.1112/jlms/s1-43.1.391 |
| [3] | Erdos, J. A.,Some questions concerning triangular operator algebras. Proc. R. In. Acad.74A (1974), 223–232. · Zbl 0291.46051 |
| [4] | Erdos, J. A.,Non-selfadjoint operator algebras. Proc. R. Irish Acad. (to appear). · Zbl 0501.47016 |
| [5] | Erdos, J. A.,On some ideals of nest algebras. Proc. London Math. Soc. (to appear). · Zbl 0523.47028 |
| [6] | Erdos, J. A. and Longstaff, W. E.,The convergence of triangular integrals of operators on Hilbert space. Indian Univ. Math. J.22 (1973), 929–938. · doi:10.1512/iumj.1973.22.22077 |
| [7] | Feintuch, A.,Strictly and strongly strictly causal linear operators.SIAM J. Math. Anal. 10 (1979), 603–613. · Zbl 0422.47025 · doi:10.1137/0510056 |
| [8] | Feintuch, A.,Strong causality and causal invertibility. SIAM J. Control and Optimization18 (1980), 317–324. · Zbl 0439.93030 · doi:10.1137/0318023 |
| [9] | Feintuch, A. and Saeks, R., System Theory: A Hilbert Space Approach. Academic, New York, 1982. · Zbl 0488.93003 |
| [10] | Gohberg, I. C. and Krein, M. G., Theory and Applications of Volterra Operators in Hilbert Space. Transi. Math. Monographs 24, American Mathematical Society, Providence, R.I., 1970. · Zbl 0194.43804 |
| [11] | Hopenwasser, A.,Completely isometric maps and triangular operator algebras. Proc. London Math. Soc. (3)25 (1972), 96–114. · Zbl 0235.46097 · doi:10.1112/plms/s3-25.1.96 |
| [12] | Kadison, R. V. and Singer, I. M.,Triangular operator algebras. Amer. J. Math.82 (1960), 227–259. · Zbl 0096.31703 · doi:10.2307/2372733 |
| [13] | Larson, D. R.,Nest algebras and similarity transformations, Annals of Mathematics (to appear). · Zbl 0606.47045 |
| [14] | Ringrose, J. R.,On some algebras of operators. Proc. London Math. Soc. (3)15 (1965), 61–83. · Zbl 0135.16804 · doi:10.1112/plms/s3-15.1.61 |
| [15] | Saeks, R.,Resolution space, operators and systems. Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 1973. · Zbl 0259.49001 |
| [16] | Zames, G.,Realizability conditions for nonlinear feedback systems. IEEE Trans. Circuit Theory11 (1964), 186–194. |
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