### Upper and lower bounds for an integral transform of positive operators in Hilbert spaces with applications

#### Abstract

\[

\mathcal{D}( w,\mu ) ( T) :=\int_{0}^{\infty }w(\lambda) (\lambda +T)^{-1} d\mu ( \lambda ) ,

\]

where the integral is assumed to exist for any positive operator \(T\) on a complex Hilbert space \(H\). In this paper we obtain several upper and lower bounds for the difference \(\mathcal{D}( w,\mu ) ( A) -\mathcal{D}( w,\mu ) ( B)\) under certain assumptions for the operators \(A\) and \(B\). Some natural applications for operator monotone and operator convex functions are also given.

#### Keywords

#### Full Text:

PDF#### References

Bhatia, R., Matrix Analysis, Springer-Verlag, New York, 1997.

Fujii, J. I., Seo, Y., On parametrized operator means dominated by power ones, Sci. Math. 1 (1998), 301–306.

Furuichi, S., Refined Young inequalities with Specht’s ratio, J. Egypt. Math. Soc. 20(1) (2011), 46–49.

Furuta, T., Concrete examples of operator monotone functions obtained by an elementary method without appealing to Lowner integral representation, Linear Algebra Appl. 429 (2008), 972–980.

Furuta, T., Precise lower bound of f(A) - f(B) for A > B > 0 and non-constant operator monotone function f on [0,infty), J. Math. Inequal. 9(1) (2015), 47–52.

Generalized Exponential Integral, Digital Library of Mathematical Functions, NIST. [Online https://dlmf.nist.gov/8.19#E1].

Heinz, E., Beitrage zur Storungsteorie der Spektralzerlegung, Math. Ann. 123 (1951), 415–438.

Incomplete Gamma and Related Functions, Definitions, Digital Library of Mathematical Functions, NIST. [Online https://dlmf.nist.gov/8.2].

Incomplete Gamma and Related Functions, Integral Representations, Digital Library of Mathematical Functions, NIST. [Online https://dlmf.nist.gov/8.6].

Lowner, K., Uber monotone MatrixFunktionen, Math. Z. 38 (1934), 177–216.

Moslehian, M. S., Najafi, H., An extension of the Lowner–Heinz inequality, Linear Algebra Appl., 437 (2012), 2359–2365.

Zuo, H., Duan, G., Some inequalities of operator monotone functions, J. Math. Inequal. 8(4) (2014), 777–781.

Zuo, H., Shi, G., Fujii, M., Refined Young inequality with Kantorovich constant, J. Math. Inequal. 5(4) (2011), 551–556.

DOI: http://dx.doi.org/10.17951/a.2022.76.1.1-15

Date of publication: 2022-10-05 20:39:29

Date of submission: 2022-10-04 18:39:31

#### Statistics

#### Indicators

### Refbacks

- There are currently no refbacks.

Copyright (c) 2022 Silvestru Sever Dragomir