  We present the application of the pseudo-spectrum method to galaxy-galaxy
lensing. We derive explicit expressions for the pseudo-spectrum analysis of the
galaxy-shear cross spectrum, which is the Fourier space counterpart of the
stacked galaxy-galaxy lensing profile. The pseudo-spectrum method corrects
observational issues such as the survey geometry, masks of bright stars and
their spikes, and inhomogeneous noise, which distort the spectrum and also mix
the E-mode and the B-mode signals. Using ray-tracing simulations in N-body
simulations including realistic masks, we confirm that the pseudo-spectrum
method successfully recovers the input galaxy-shear cross spectrum. We also
investigate the covariance of the galaxy-shear cross spectrum using the
ray-tracing simulations to show that there is an excess covariance relative to
the Gaussian covariance at small scales where the shot noise is dominated in
the Gaussian approximation. We find that the excess of the covariance is
consistent with the expectation from the halo sample variance (HSV), which
originates from the matter fluctuations at scales larger than the survey area.
We apply the pseudo-spectrum method to the observational data of
Canada-France-Hawaii Telescope Lensing survey (CFHTLenS) shear catalogue and
three different spectroscopic samples of Sloan Digital Sky Survey Luminous Red
Galaxy (SDSS LRG), and Baryon Oscillation Spectroscopic Survey (BOSS) CMASS and
LOWZ galaxies. The galaxy-shear cross spectra are significantly detected at the
level of 7-10$\sigma$ using the analytic covariance with the HSV contribution
included. We also confirm that the observed spectra are consistent with the
halo model predictions with the halo occupation distribution parameters
estimated from previous work. This work demonstrates the viability of
galaxy-galaxy lensing analysis in the Fourier space.
