Polyak's theorem on Hilbert spaces

Abstract

We extend to infinite dimensional Hilbert spaces a celebrated result, due to B. Polyak, about the convexity of the joint image of quadratic functions. We show sufficient conditions which assure that the joint image is also closed. However, we prove that the closedness part of Polyak's theorem does not hold in general in the infinite dimensional setting. Finally, we give some applications to S-lemma type results.

Keywords:

• Quadratic forms on Hilbert spaces
• convexity
• numerical range
• S-lemma

AMS 2010:

• 47A07
• 47A12
• 46C05
• 52A10
• 52A15
• 90C20

Acknowledgments

We thank the anonymous referee for carefully reading the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Maximiliano Contino was supported by CONICET-PIP 11220200102127CO and by María Zambrano Postdoctoral Grant CT33/21 at Universidad Complutense de Madrid financed by the Ministry of Universities with Next Generation EU funds. Guillermina Fongi was supported by ANPCyT-PICT 2017-0883 and ING-586-UNR. Santiago Muro was supported by ANPCyT-PICT 2018-04250, ING-586-UNR and CONICET-PIP 11220130100329CO.