Numerical radius inequalities and estimation of zeros of polynomials
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914910474076160 |
|---|---|
| author | Jana, Suvendu Bhunia, Pintu Paul, Kallol |
| author_facet | Jana, Suvendu Bhunia, Pintu Paul, Kallol |
| contents | Let $A$ be a bounded linear operator defined on a complex Hilbert space and let $|A|=(A^*A)^{1/2}$ be the positive square root of $A$.
Among other refinements of the well known numerical radius inequality $w^2(A)\leq \frac12 \|A^*A+AA^*\|$,
we show that
\begin{eqnarray*}
w^2(A)&\leq&\frac{1}{4} w^2 \left(|A|+i|A^*|\right)+\frac{1}{8}\left\||A|^2+|A^*|^2\right \|+\frac{1}{4}w\left(|A||A^*|\right)
&\leq& \frac12 \|A^*A+AA^*\|.
\end{eqnarray*} Also, we develop inequalities involving numerical radius and spectral radius for the sum of the product operators, from which we derive the following inequalities $$ w^p(A) \leq \frac{1}{\sqrt{2} } w(|A|^p+i|A^*|^p )\leq \|A\|^p$$ for all $p\geq 1.$ Further, we derive new bounds for the zeros of complex polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_03159 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Numerical radius inequalities and estimation of zeros of polynomials Jana, Suvendu Bhunia, Pintu Paul, Kallol Functional Analysis Primary 47A12, 26C10 Secondary 47A30, 30C15 Let $A$ be a bounded linear operator defined on a complex Hilbert space and let $|A|=(A^*A)^{1/2}$ be the positive square root of $A$. Among other refinements of the well known numerical radius inequality $w^2(A)\leq \frac12 \|A^*A+AA^*\|$, we show that \begin{eqnarray*} w^2(A)&\leq&\frac{1}{4} w^2 \left(|A|+i|A^*|\right)+\frac{1}{8}\left\||A|^2+|A^*|^2\right \|+\frac{1}{4}w\left(|A||A^*|\right) &\leq& \frac12 \|A^*A+AA^*\|. \end{eqnarray*} Also, we develop inequalities involving numerical radius and spectral radius for the sum of the product operators, from which we derive the following inequalities $$ w^p(A) \leq \frac{1}{\sqrt{2} } w(|A|^p+i|A^*|^p )\leq \|A\|^p$$ for all $p\geq 1.$ Further, we derive new bounds for the zeros of complex polynomials. |
| title | Numerical radius inequalities and estimation of zeros of polynomials |
| topic | Functional Analysis Primary 47A12, 26C10 Secondary 47A30, 30C15 |
| url | https://arxiv.org/abs/2301.03159 |