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.

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Bhunia, P., Feki, K. & Paul, K. Correction to: A-Numerical Radius Orthogonality and Parallelism of Semi-Hilbertian Space Operators and Their Applications. Bull. Iran. Math. Soc. (2020). https://doi.org/10.1007/s41980-020-00412-7

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