Good evening. Note: $||x||=|f|^{2}+|g|^{2}=1$
I have a question about notation mainly and about evaluation of something. In one paper we have a matrix $$M = \begin{pmatrix} 0 & 2& \\ 1 & 1&\\ \end{pmatrix}$$ $$N= \begin{pmatrix} 1 & 1 & \\ 3 & 0 & \\ \end{pmatrix} $$ In the article I am reading, it is said that $w(M^{*}N)=3$, where $w=\sup\{|\langle Tx,x\rangle||x\in H,||x||=1\}$ Also, $$|| |N|^{2}+|M|^{2} ||=15.3007$$ My first question is, how does one find that $w(M^{*}N)=3$, also i am confused as to what does $|N|$ mean? Please write a definition and step by step evaluation of both cases if possible. Here is my try $$\begin{pmatrix} 0 & 1 & \\ 2 & 1 & \\ \end{pmatrix} \times \begin{pmatrix} 1 & 1 & \\ 3 & 0 & \\ \end{pmatrix} =\begin{pmatrix} 3 & 0 & \\ 5 & 2 & \\ \end{pmatrix} $$ Now when we look for $M^{*}Nx$ we get $$= \begin{pmatrix} 3 & 0 & \\ 5 & 2 & \\ \end{pmatrix} \times \begin{pmatrix} f & \\ g & \\ \end{pmatrix} =(3f,5f+2g) $$ Which when we take dot product of with $x=(f,g)$ we get $\langle (3f,5f+2g),(f,g)\rangle = 3f^{2}+5fg+2g^{2}$ Now taking absolute value, we get $$|\langle M^{*}Nx,x\rangle|\leq 3|f^{2}|+5|fg|+2|g^{2}|\leq 3|f^{2}|+2|g^{2}|+5\frac{|f|^{2}+|g|^{2}}{2}=5+3|f^{2}|+2|g^{2}|.$$ I do not know how to proceed here to obtain 3 as the upper bound for the numerical radius. Also, I have no idea what $|N|$ is for a matrix. Also what $||N||$ is, I have no clue. Thank you for answering and reading this!