# Correction to: A-Numerical Radius Orthogonality and Parallelism of Semi-Hilbertian Space Operators and Their Applications

## Correction to: Bulletin of the Iranian Mathematical Society https://doi.org/10.1007/s41980-020-00392-8

In the original article published, during the final typesetting stage the equation in the proof of the Theorem 3.5 was published incorrectly. The correct equation is:

### Theorem 3.5

Let $$T,S\in {\mathcal {B}}_A({\mathcal {H}})$$ be such that $$T \parallel _A S$$. Also let $${\mathbb {A}}=\left( \begin{array}{cc} A&{}0 \\ 0&{}A \end{array}\right)$$. Then

\begin{aligned} \omega _{{\mathbb {A}}}\left( \begin{array}{cc} O&{}T \\ e^{-2\mathrm{{i}}\beta }S^{\sharp _A}&{}O \end{array}\right) = \tfrac{1}{2}(\Vert T\Vert _A+\Vert S\Vert _A), \end{aligned}

for some real $$\beta$$.

### Proof

We have $$\Vert T+ e^{2\mathrm{{i}}\beta }S\Vert _A=\Vert T\Vert _A+\Vert S\Vert _A$$ for some real $$\beta$$, as $$T \Vert _A S$$. Therefore, using Theorem 3.4, we have

\begin{aligned} \Vert T\Vert _A+\Vert S\Vert _A&= \Vert T+ e^{2\mathrm{{i}}\beta }S\Vert _A\\&\le 2 \omega _{{\mathbb {A}}}\left( \begin{array}{cc} O&{}T \\ e^{-2\mathrm{{i}}\beta }S^{\sharp _A}&{}O \end{array}\right) \\&\le \Vert T\Vert _A+\Vert S\Vert _A. \end{aligned}

It follows that

\begin{aligned} \omega _{{\mathbb {A}}}\left( \begin{array}{cc} O&{}T \\ e^{-2\mathrm{{i}}\beta }S^{\sharp _A}&{}O \end{array}\right) = \tfrac{1}{2}(\Vert T\Vert _A+\Vert S\Vert _A), \end{aligned}

for some real $$\beta$$.

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Correspondence to Kais Feki.