Some numerical radius inequality for several operators

Let T∈B(H) be a bounded linear operator and ω(T) denotes the numerical radius of T. Let S1,…,Sm,T1,…,Tm∈B(H) such that SiTj=TjSi and Si⁎Tj=TjSi⁎ for all 1≤i,j,≤m. It is shown thatω(∑i=1mSiTi)≤12‖∑i=1mSi⁎Si+SiSi⁎‖12‖∑i=1mTi⁎Ti+TiTi⁎‖12. Moreover, we show thatω(∑i=1mSiTi+TiSi)≤‖∑i=1mSi⁎Si+SiSi⁎‖12‖∑i=1mTi⁎Ti+TiTi⁎‖12, for all S1,…,Sm,T1,…,Tm∈B(H). In particular, for m=1, we refine some well known inequalities.