Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces

Aimed toward researchers, postgraduate students, and scientists in linear operator theory and mathematical inequalities, this self-contained monograph focuses on numerical radius inequalities for bounded linear operators on complex Hilbert spaces for the case of one and two operators. Students at the graduate level will learn some essentials that may be useful for reference in courses in functional analysis, operator theory, differential equations, and quantum computation, to name several. Chapter 1 presents fundamental facts about the numerical range and the numerical radius of bounded linear operators in Hilbert spaces. Chapter 2 illustrates recent results obtained concerning numerical radius and norm inequalities for one operator on a complex Hilbert space, as well as some special vector inequalities in inner product spaces due to Buzano, Goldstein, Ryff and Clarke as well as some reverse Schwarz inequalities and Grüss type inequalities obtained by the author. Chapter 3 presents  recent results regarding the norms and the numerical radii of two bounded linear operators. The techniques shown in this chapter are elementary but elegant and may be accessible to undergraduate students with a working knowledge of operator theory. A number of vector inequalities in inner product spaces as well as inequalities for means of nonnegative real numbers are also employed in this chapter. All the results presented are completely proved and the original references are mentioned.

• Euclidean operator radius
• Grüss type inequalities
• Linear operator theory
• complex Hilbert spaces
• numerical radius inequalities
• ordinary differential equations

From the book reviews:

“The aim of this book is to provide several inequalities, mainly obtained by the author, concerning the numerical radius of linear operators. … The book is easy to read and should be accessible to undergraduates taking a course in operator theory.” (Cătălin Badea, zbMATH, Vol. 1302, 2015)

“The author discusses various numerical radius inequalities for bounded linear operators in complex Hilbert spaces. … The book is appropriate for researchers and graduate students in the area of linear operator theory in Hilbert spaces, or as a reference book for researchers in different mathematical disciplines using inequalities involving the numerical radius of a linear operator. … the book is well written and provides a good summary of the author’s recent results.” (Tsvetanka Sendova, Mathematical Reviews, June, 2014)

• College of Engineering & Science; School of Computational and Applied Mathematics, Victoria University, Melbourne, Australia, and University of the Withwatersrand, Braamfontein, South Africa

  Silvestru Sever Dragomir

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