Estimations of Euclidean operator radius

We develop several Euclidean operator radius bounds for the product of two $d$-tuple operators using positivity criteria of a $2\times 2$ block matrix whose entries are $d$-tuple operators. From these bounds, by using the polar decomposition of operators, we obtain Euclidean operator radius bounds for $d$-tuple operators. Among many other interesting bounds, it is shown that \begin{eqnarray*} w_e(\mathbf{A}) &\leq&\frac1{\sqrt2} \mathbf{A}\|^{1/2}\sqrt{\left\|\sum_{k=1}^{d} (|A_k|+|A_k^*|)\right\|}, \end{eqnarray*} where $w_e(\mathbf{A})$ and $\|\mathbf{A}\|$ are the Euclidean operator radius and the Euclidean operator norm, respectively, of a $d$-tuple operator $\mathbf{A}=(A_1,A_2, \ldots,A_d).$ Further, we develop an upper bound for the Euclidean operator radius of $n\times n$ operator matrix whose entries are $d$-tuple operators. In particular, it is proved that if $\begin{bmatrix} \mathbf{A_{ij}} \end{bmatrix}_{n\times n}$ is an $n\times n$ operator matrix then $$w_e\left( \begin{bmatrix} \mathbf{A_{ij}} \end{bmatrix}_{n\times n}\right)\leq w \left(\begin{bmatrix} a_{ij} \end{bmatrix}_{n\times n}\right),$$ where each $\mathbf{A_{ij}}$ is a $d$-tuple operator, $1\leq i,j\leq n$, $a_{ij}=w_e(\mathbf{A_{ij}})\, \textit{ if i=j}$, a_{ij}= \sqrt{w_e\left(|\mathbf{A_{ji}|}+|\mathbf{A_{ij}^*}|\right)w_e\left(|\mathbf{A_{ij}|}+|\mathbf{A_{ji}^*}|\right)}\,\textit{ if i<j}, and a_{ij}= 0\,\textit{ if i>j}. Other related applications are also discussed.