Attenuation compensation (AC), while being beneficial for visual-interpretation tasks in myocardial perfusion imaging (MPI) by SPECT, typically requires the availability of a separate X-ray CT component, leading to additional radiation dose, higher costs, and potentially inaccurate diagnosis due to SPECT/CT misalignment. To address these issues, we developed a method for cardiac SPECT AC using deep learning and emission scatter-window photons without a separate transmission scan (CTLESS). In this method, an estimated attenuation map reconstructed from scatter-energy window projections is segmented into different regions using a multi-channel input multi-decoder network trained on CT scans. Pre-defined attenuation coefficients are assigned to these regions, yielding the attenuation map used for AC. We objectively evaluated this method in a retrospective study with anonymized clinical SPECT/CT stress MPI images on the clinical task of detecting defects with an anthropomorphic model observer. CTLESS yielded statistically non-inferior performance compared to a CT-based AC (CTAC) method and significantly outperformed a non-AC (NAC) method on this clinical task. Similar results were observed in stratified analyses with different sexes, defect extents and severities. The method was observed to generalize across two SPECT scanners, each with a different camera. In addition, CTLESS yielded similar performance as CTAC and outperformed NAC method on the metrics of root mean squared error and structural similarity index measure. Moreover, as we reduced the training dataset size, CTLESS yielded relatively stable AUC values and generally outperformed another DL-based AC method that directly estimated the attenuation coefficient within each voxel. These results demonstrate the capability of the CTLESS method for transmission-less AC in SPECT and motivate further clinical evaluation.