[*2010 Mathematics Subject Classification*]{}: Primary 62M10; Secondary 62M15, 60G10\
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[*Keywords and Phrases*]{}: Bootstrap, Functional Time Series, Hypothesis Testing, Spectral Density Operator\
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[*Short title*]{}: Testing Equality of Spectral Density Operatorsversion:

Introduction {#S1}
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Functional time series form a natural class of functional data occurred in many applications such as daily curves of financial transactions, daily images of geophysical and environmental data and daily curves of temperature measurements. Such curves or images are viewed as functions in appropriate spaces since an observed intensity is available at each point on a line segment, a portion of a plane or a volume. Moreover, and most importantly, such functional time series exhibit temporal dependence while ignoring this dependence may result in misleading conclusions and inferential procedures.

Comparing characteristics of two or more groups of functional data forms an important problem of statistical inference with a variety of applications. For instance, comparing the mean functions between independent groups of independent and identically distributed (i.i.d.) functional data or of functional time series has attracted considerable interest in the literature, see, e.g., Benko [*et al.*]{} (2009), Zhang [*et al.*]{} (2010), Horváth and Kokoszka (2012, Chapter 5), Horváth [*et al.*]{} (2013) and Paparoditis and Sapatinas (2016). In 