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An operator T ∈ B(H) is said to be (p, k)-quasiposinormal operator, if T * k (c 2 (T * T ) p − (T T * ) p )T k ≥ 0 for a positive integer 0 < p ≤ 1, some c > 0 and a positive integer k. In this paper, we prove that, the (p, k) quasi-posinormal operator is a pole of resolvent of T * . Then we prove that if {T n } is a sequence of operators in the class (p, k) − Q and (p, k) − QP which converges in the operator norm topology to an operator T in the same class, then the functions spectrum, Weyl spectrum, Browder spectrum and essential surjectivity spectrum are continuous at T .

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This study explores the dynamic nature of return, volatility and correlation transmission mechanism among Indian exchange rates relative to US dollar (USD), Great Britain pound (GBP), euro and Japanese yen. We make use of the dynamic conditional correlation (DCC) vector autoregressive multivariate generalized autoregressive conditional heteroskedasticity (VAR (1)-MVGARCH) model which is capable of capturing the interactive dynamics in the first moment and the second moment of the time series. Our empirical results point to a significant unidirectional return spillover from euro and Japanese yen to USD and bidirectional return spillover between GBP and Japanese yen. We also find evidence of significant volatility spillover effect from USD to GBP, euro and Japanese Yen and from GBP and Euro to USD. Moreover, we find that the time-varying conditional correlations between exchange rate changes dynamically over time and are widely distributed with higher volatility during the period of global financial crisis for all USD and other exchange rate pairs.