### Closedness and invertibility for the sum of two closed operators

Nikolaos Roidos

#### Abstract

We show a Kalton–Weis type theorem for the general case of noncommuting operators. More precisely, we consider sums of two possibly noncommuting linear operators defined in a Banach space such that one of the operators admits a bounded $H^\infty$-calculus, the resolvent of the other one satisfies some weaker boundedness condition and the commutator of their resolvents has certain decay behavior with respect to the spectral parameters. Under this consideration, we show that the sum is closed and that after a sufficiently large positive shift it becomes invertible and moreover sectorial. As an application we recover a classical result on the existence, uniqueness, and maximal $L^{p}$-regularity for solutions of the abstract linear nonautonomous parabolic problem.

#### Article information

Source
Adv. Oper. Theory Volume 3, Number 3 (2018), 582-605.

Dates
Accepted: 8 February 2018
First available in Project Euclid: 3 March 2018

https://projecteuclid.org/euclid.aot/1520046046

Digital Object Identifier
doi:10.15352/aot.1801-1297

#### Citation

Roidos, Nikolaos. Closedness and invertibility for the sum of two closed operators. Adv. Oper. Theory 3 (2018), no. 3, 582--605. doi:10.15352/aot.1801-1297. https://projecteuclid.org/euclid.aot/1520046046

#### References

• H. Amann, Linear and quasilinear parabolic problems Vol. I, Abstract linear theory. Monographs in Mathematics, 89, Birkhäuser Boston, Inc., Boston, MA, 1995.
• H. Amann, Maximal regularity for nonautonomous evolution equations, Adv. Nonlinear Stud. 4 (2004), no. 4, 417–430.
• W. Arendt, R. Chill, S. Fornaro, and C. Poupaud, $L^{p}$-maximal regularity for non-autonomous evolution equations, J. Differential Equations 237 (2007), no. 1, 1–26.
• G. Da Prato and P. Grisvard, Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl. (9) 54 (1975), no. 3, 305–387.
• R. Denk, M. Hieber, and J. Prüss, $R$-boundedness, Fourier multipliers, and problems of elliptic and parabolic type, Mem. Amer. Math. Soc. 166 (2003), no. 788, viii+114 pp.
• G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196 (1987), no. 2, 189–201.
• M. Haase, The functional calculus for sectorial operators, Operator theory: Advances and applications, \bf169, Birkhäuser Verlag, Basel, 2006.
• N. J. Kalton and L. Weis, The $H^{\infty}$-calculus and sums of closed operators, Math. Ann. 321 (2001), no. 2, 319–345.
• P. C. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^{\infty}$-functional calculus, Functional Analytic Methods for Evolution Equations, 65–311, Lecture Notes in Math., 1855, Springer, Berlin, 2004.
• S. Monniaux and J. Prüss, A theorem of the Dore-Venni type for noncommuting operators, Trans. Amer. Math. Soc. 349 (1997), no. 12, 4787–4814.
• J. Prüss and R. Schnaubelt, Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time, J. Math. Anal. Appl. \bf256 (2001), no. 2, 405–430.
• J. Prüss and G. Simonett, $H^\infty$-calculus for the sum of non-commuting operators, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3549–3565.
• N. Roidos, On the inverse of the sum of two sectorial operators, J. Funct. Anal. \bf265 (2013), no. 2, 208–222.
• N. Roidos, Preserving closedness of operators under summation, J. Funct. Anal. \bf266 (2014), no. 12, 6938–6953.
• N. Roidos and E. Schrohe, Smoothness and long time existence for solutions of the porous medium equation on manifolds with conical singularities, arXiv:1708.07542.