# Boundedness properties of resolvents and semigroups of operators

Banach Center Publications (1997)

- Volume: 38, Issue: 1, page 59-74
- ISSN: 0137-6934

## Access Full Article

top## Abstract

top## How to cite

topvan Casteren, J.. "Boundedness properties of resolvents and semigroups of operators." Banach Center Publications 38.1 (1997): 59-74. <http://eudml.org/doc/208649>.

@article{vanCasteren1997,

abstract = {Let T: H → H be an operator in the complex Hilbert space H. Suppose that T is square bounded in average in the sense that there exists a constant M(T) with the property that, for all natural numbers n and for all x ∈ H, the inequality $1/(n+1) ∑_\{j=0\}^n ∥T^\{j\}x∥^2 ≤ M(T)^\{2\} ∥x∥^\{2\}$ is satisfied. Also suppose that the adjoint T* of the operator T is square bounded in average with constant M(T*). Then the operator T is power bounded in the sense that $sup\{∥T^i\{n\}∥: n ∈ ℕ\}$ is finite. In fact the following inequality is valid for all n ∈ ℕ: ∥Tn∥ ≤ e M(T)M(T*). Suppose that T has a bounded everywhere defined inverse S with the property that for λ in the open unit disc of ℂ the operator $(I-λS)^\{-1\}$ exists and that the expression $sup\{(1-|λ|)∥(I - λS)^\{-1\}∥: |λ| <1\}$ is finite. If T is power bounded, then so is S and hence in such a situation the operator T is similar to a unitary operatorsimilarity to unitary operator. If both the operators T* and S are square bounded in average, then again the operator T is similar to a unitary operator. Similar results hold for strongly continuous semigroups instead of (powers) of a single operator. Some results are also given in the more general Banach space context.
},

author = {van Casteren, J.},

journal = {Banach Center Publications},

keywords = {operator Poisson kernel; bounded semigroup; power bounded operator; square bounded in average; square bounded average; power bounded; similar to a unitary operator; strongly continuous semigroups},

language = {eng},

number = {1},

pages = {59-74},

title = {Boundedness properties of resolvents and semigroups of operators},

url = {http://eudml.org/doc/208649},

volume = {38},

year = {1997},

}

TY - JOUR

AU - van Casteren, J.

TI - Boundedness properties of resolvents and semigroups of operators

JO - Banach Center Publications

PY - 1997

VL - 38

IS - 1

SP - 59

EP - 74

AB - Let T: H → H be an operator in the complex Hilbert space H. Suppose that T is square bounded in average in the sense that there exists a constant M(T) with the property that, for all natural numbers n and for all x ∈ H, the inequality $1/(n+1) ∑_{j=0}^n ∥T^{j}x∥^2 ≤ M(T)^{2} ∥x∥^{2}$ is satisfied. Also suppose that the adjoint T* of the operator T is square bounded in average with constant M(T*). Then the operator T is power bounded in the sense that $sup{∥T^i{n}∥: n ∈ ℕ}$ is finite. In fact the following inequality is valid for all n ∈ ℕ: ∥Tn∥ ≤ e M(T)M(T*). Suppose that T has a bounded everywhere defined inverse S with the property that for λ in the open unit disc of ℂ the operator $(I-λS)^{-1}$ exists and that the expression $sup{(1-|λ|)∥(I - λS)^{-1}∥: |λ| <1}$ is finite. If T is power bounded, then so is S and hence in such a situation the operator T is similar to a unitary operatorsimilarity to unitary operator. If both the operators T* and S are square bounded in average, then again the operator T is similar to a unitary operator. Similar results hold for strongly continuous semigroups instead of (powers) of a single operator. Some results are also given in the more general Banach space context.

LA - eng

KW - operator Poisson kernel; bounded semigroup; power bounded operator; square bounded in average; square bounded average; power bounded; similar to a unitary operator; strongly continuous semigroups

UR - http://eudml.org/doc/208649

ER -

## References

top- [1] J. M. Anderson, J. G. Clunie and Ch. Pommerenke, On Bloch functions and normal families, J. Reine Angew. Math. 270 (1974), 12-37.
- [2] J. M. Anderson and A. L. Shields, Coefficient multipliers of Bloch functions, Trans. Amer. Math. Soc. 224 (1976), 255-265. Zbl0352.30032
- [3] G. Bennett, D. A. Stegenga and R. M. Timoney, Coefficients of Bloch functions and Lipschitz functions, Illinois J. Math. 25 (1981), 520-531. Zbl0443.30041
- [4] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973. Zbl0271.46039
- [5] F. F. Bonsall and J. Duncan, Numerical Ranges II, London Math. Soc. Lecture Note Ser. 10, Cambridge University Press, 1973. Zbl0262.47001
- [6] P. Duren, Theory of ${H}^{p}$-spaces, Academic Press, New York, 1970. Zbl0215.20203
- [7] R. E. Edwards and G. I. Gaudry, Littlewood-Paley and Multiplier Theory, Springer, Berlin, 1977.
- [8] H. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer, Berlin, 1974. Zbl0837.43002
- [9] H. G. Garnir, K. R. Unni and J. H. Williams, Functional Analysis and its Applications, Lecture Notes in Math. 399, Springer, Berlin, 1974.
- [10] I. C. Gohberg and M. G. Krein, On a description of contraction operators similar to unitary ones, Funktsional. Anal. i Prilozhen. 1 (1) (1976), 38-60 (in Russian).
- [11] P. R. Halmos, A Hilbert Space Problem Book, Van Nostrand, Princeton, N.J., 1976.
- [12] W. K. Hayman, Multivalent Functions, Cambridge University Press, Cambridge, 1958.
- [13] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc., Providence R.I., 1957.
- [14] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1976. Zbl0342.47009
- [15] M. G. Krein, Analytic problems in the theory of linear operators in Hilbert space (abstract), Amer. Math. Soc. Transl. 70 (1968), 68-72.
- [16] M. G. Krein, Analytic problems in the theory of linear operators in Hilbert space, in: Proc. Internat. Congress Math. Moscow 1966, Mir, Moscow, 1968, 189-216 (in Russian).
- [17] M. Mbekhta and J. Zemánek, Sur le théorème ergodique uniforme et le spectre, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 1155-1158.
- [18] W. Rudin, Real and Complex Analysis, 2nd ed., McGraw-Hill, New York, 1974. Zbl0278.26001
- [19] A. L. Shields, On Möbius bounded operators, Acta Sci. Math. (Szeged) 40 (1978), 371-374. Zbl0358.47025
- [20] J. G. Stampfli, A local spectral theory for operators, III: Resolvents, spectral sets and similarity, Trans. Amer. Math. Soc. 168 (1972), 133-151. Zbl0245.47002
- [21] E. M. Stein, Singular Integral Operators and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University Press, 1970.
- [22] J. C. Strikwerda, Finite Differences and Partial Differential Equations, The Wadsworth and Brooks/Cole Math. Series vol. 1989 (1), Wadsworth & Brooks, Pacific Grove, Calif., 1989.
- [23] B. Sz.-Nagy, On uniformly bounded linear transformations in Hilbert space, Acta Sci. Math. (Szeged) 11 (1947), 152-157.
- [24] B. Sz.-Nagy and C. Foiaş, Sur les contractions de l'espace de Hilbert X; contractions similaires à des transformations unitaires, ibid. 26 (1965), 79-91. Zbl0138.38905
- [25] B. Sz.-Nagy and C. Foiaş, Analyse Harmonique des Opérateurs de l'Espace de Hilbert, Akadémiai Kiadó, Budapest, 1967. Zbl0157.43201
- [26] J. A. van Casteren, A problem of Sz.-Nagy, Acta Sci. Math. (Szeged) 42 (1980), 189-194. Zbl0437.47001
- [27] J. A. van Casteren, Operators similar to unitary or self-adjoint ones, Pacific J. Math. 104 (1983), 241-255. Zbl0457.47002
- [28] J. A. van Casteren, Generators of Strongly Continuous Semigroups, Pitman, London, 1985.
- [29] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1972. Zbl0060.24801
- [30] D. V. Widder, An Introduction to Transform Theory, Academic Press, New York, 1971. Zbl0219.44001
- [31] K. Yosida, Functional Analysis, 3rd ed., Springer, Berlin, 1971.
- [32] J. Zemánek, On the Gelfand-Hille theorems, in: Functional Analysis and Operator Theory, Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., 1994, 369-385. Zbl0822.47005

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.